From bd9c46d3e4c72e48cbf7c1ccf6439e558991ec21 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 19 May 2026 23:35:01 -0400 Subject: [PATCH] Add level resolutions of maximal planar graphs paper Migrate the paper content into the amsart template and include the supporting experiments scripts. --- .../experiments/README.md | 58 ++ .../experiments/balanced_layout.py | 212 +++++ .../experiments/coverage.py | 85 ++ .../experiments/coverage_chunked.py | 99 ++ .../experiments/coverage_fast.py | 78 ++ .../experiments/coverage_new_def.py | 104 +++ .../experiments/face_counting.py | 125 +++ .../experiments/four_color.py | 98 ++ .../experiments/four_color_viz.py | 256 ++++++ .../experiments/level_cycles.py | 173 ++++ .../experiments/n11_chunk.py | 57 ++ .../experiments/n7_examples.py | 251 ++++++ .../experiments/orbit_check.py | 123 +++ .../experiments/outerplanar_check.py | 55 ++ .../experiments/plot_oct.py | 104 +++ .../experiments/triangulation_gen.py | 78 ++ .../experiments/triangulation_gen_fast.py | 100 +++ .../paper-old.aux | 48 + .../paper-old.fdb_latexmk | 178 ++++ .../paper-old.fls | 846 ++++++++++++++++++ .../paper-old.log | 586 ++++++++++++ .../paper-old.out | 10 + .../paper-old.pdf | Bin 0 -> 211709 bytes .../paper-old.tex | 361 ++++++++ .../paper.aux | 54 ++ .../paper.fdb_latexmk | 93 ++ .../paper.fls | 508 +++++++++++ .../paper.log | 321 +++++++ .../paper.out | 11 + .../paper.pdf | Bin 0 -> 235833 bytes .../paper.tex | 420 +++++++++ 31 files changed, 5492 insertions(+) create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/README.md create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/balanced_layout.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_chunked.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_fast.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_new_def.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/face_counting.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/four_color.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/four_color_viz.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/level_cycles.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/n11_chunk.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/n7_examples.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/orbit_check.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/outerplanar_check.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/plot_oct.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/triangulation_gen.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/experiments/triangulation_gen_fast.py create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper-old.aux create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper-old.fdb_latexmk create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper-old.fls create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper-old.log create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper-old.out create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper-old.pdf create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper-old.tex create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper.aux create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper.fdb_latexmk create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper.fls create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper.log create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper.out create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper.pdf create mode 100644 papers/level_resolutions_of_maximal_planar_graphs/paper.tex diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/README.md b/papers/level_resolutions_of_maximal_planar_graphs/experiments/README.md new file mode 100644 index 0000000..fb8a193 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/README.md @@ -0,0 +1,58 @@ +# Level Resolution Experiments + +Computational investigation of a structural proof strategy for the four +color theorem via *level resolutions* of maximal planar graphs. + +See `paper.tex` for full definitions, conjectures, and findings. + +## Files + +### Core library +- `level_cycles.py` — levels, level subgraphs, level cycles, resolution + enumeration (used by old-definition coverage). +- `triangulation_gen.py` — vertex-insertion + flip closure (good to n=10). +- `triangulation_gen_fast.py` — WL-hash pre-filter for n ≥ 11. +- `balanced_layout.py` — Tutte-init random-search planar layout. +- `four_color.py` — level 4-coloring via parity 2-coloring of L_k. + +### Experiments +- `coverage_new_def.py` — **coverage under the cleaner definition**: + G' is a level resolution of G via S iff its parity subgraphs are + bipartite. Reachability reduces to "G' admits a bipartite 2-partition + with cardinality matching some BFS-realizable parity split." +- `coverage.py`, `coverage_fast.py`, `coverage_chunked.py` — coverage + under the OLD (stricter) definition involving specific edge flips on + level cycles. +- `face_counting.py` — per-target preimage counts (N_iso, N_paths) under + the old definition. +- `orbit_check.py` — orbit-counting with k-flip reverse-preimages (used + for old-definition icosahedron analysis). + +### Visualizations +- `plot_oct.py`, `n7_examples.py`, `four_color_viz.py`. + +## Summary under the new definition + +| n | iso-classes | reachable | md4 reachable | +|----|-------------|-----------|---------------| +| 6 | 2 | 2 | 1/1 | +| 7 | 5 | 5 | 1/1 | +| 8 | 14 | 14 | 2/2 | +| 9 | 50 | 50 | 5/5 | +| 10 | 233 | 233 | 12/12 | +| 11 | 1249 | 1249 | 34/34 | +| 12 | icosahedron | reachable | yes | + +**Every iso-class is reachable** at every tested size. The previously +"uncovered" classes T1 (n=7) and T6 (n=8) under the old definition are +both reachable under the cleaner definition. + +The new definition makes coverage equivalent to 4CT plus a BFS-realizable +partition cardinality constraint, raising the question of what additional +structure on the preimage G would make the framework non-circular. + +## Dependencies + +``` +pip install networkx matplotlib numpy scipy +``` diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/balanced_layout.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/balanced_layout.py new file mode 100644 index 0000000..0b02d35 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/balanced_layout.py @@ -0,0 +1,212 @@ +""" +Face-area-balanced planar layout for maximal planar graphs. + +Starts from a Tutte embedding (outer face on a circle) and uses random-search +optimization to equalize interior face areas while maintaining planarity. +""" +import networkx as nx +import numpy as np +import scipy.linalg + + +def _get_all_faces(emb): + """Enumerate all faces of a PlanarEmbedding.""" + seen, faces = set(), [] + for v in emb.nodes(): + for w in emb[v]: + if (v, w) not in seen: + face = emb.traverse_face(v, w) + for i in range(len(face)): + seen.add((face[i], face[(i+1) % len(face)])) + faces.append(tuple(face)) + return faces + + +def _tutte_layout(G, outer_face): + """Standard uniform Tutte embedding with outer face on unit circle.""" + outer = list(outer_face) + k = len(outer) + fixed_pos = {v: np.array([np.cos(2*np.pi*i/k - np.pi/2), + np.sin(2*np.pi*i/k - np.pi/2)]) + for i, v in enumerate(outer)} + interior = [v for v in G.nodes() if v not in fixed_pos] + if not interior: + return fixed_pos + int_idx = {v: i for i, v in enumerate(interior)} + m = len(interior) + A = np.zeros((m, m)); bx = np.zeros(m); by = np.zeros(m) + for v in interior: + i = int_idx[v]; nbrs = list(G.neighbors(v)); deg = len(nbrs) + A[i, i] = -1.0 + for u in nbrs: + if u in int_idx: + A[i, int_idx[u]] += 1.0 / deg + else: + bx[i] -= fixed_pos[u][0] / deg + by[i] -= fixed_pos[u][1] / deg + px = scipy.linalg.solve(A, bx) + py = scipy.linalg.solve(A, by) + pos = dict(fixed_pos) + for v in interior: + pos[v] = np.array([px[int_idx[v]], py[int_idx[v]]]) + return pos + + +def _segments_cross(p1, p2, p3, p4): + """True iff open segments p1-p2 and p3-p4 cross (not at endpoints).""" + def cross(o, a, b): + return (a[0]-o[0])*(b[1]-o[1]) - (a[1]-o[1])*(b[0]-o[0]) + d1 = cross(p3, p4, p1); d2 = cross(p3, p4, p2) + d3 = cross(p1, p2, p3); d4 = cross(p1, p2, p4) + return ((d1 > 0 and d2 < 0) or (d1 < 0 and d2 > 0)) and \ + ((d3 > 0 and d4 < 0) or (d3 < 0 and d4 > 0)) + + +def _count_crossings(G, pos): + """Number of pairs of edges whose straight-line segments cross.""" + edges = list(G.edges()) + n = 0 + for i in range(len(edges)): + a, b = edges[i] + for j in range(i+1, len(edges)): + c, d = edges[j] + if len({a, b, c, d}) < 4: + continue + if _segments_cross(pos[a], pos[b], pos[c], pos[d]): + n += 1 + return n + + +def _face_area(pos, face): + """Unsigned area of a triangular (or polygonal) face.""" + pts = [pos[v] for v in face] + return 0.5 * abs((pts[1][0]-pts[0][0])*(pts[2][1]-pts[0][1]) + - (pts[1][1]-pts[0][1])*(pts[2][0]-pts[0][0])) + + +def _edge_lengths(G, pos): + return [np.linalg.norm(pos[u] - pos[v]) for u, v in G.edges()] + + +def _score(G, pos, interior_faces, w_area=1.0, w_edge=0.3): + """Combined penalty: normalized variance of face areas + edge lengths.""" + areas = np.array([_face_area(pos, f) for f in interior_faces]) + a_mean = areas.mean() + area_score = np.sum((areas - a_mean)**2) / (a_mean**2) + lens = np.array(_edge_lengths(G, pos)) + l_mean = lens.mean() + edge_score = np.sum((lens - l_mean)**2) / (l_mean**2) + return w_area * area_score + w_edge * edge_score + + +def balanced_planar_layout( + G, + outer_face, + n_explore=8000, + n_refine=4000, + explore_step=0.30, + refine_step=0.05, + w_area=1.0, + w_edge=0.3, + seed=42, + verbose=False, +): + """ + Compute a planar layout for a maximal planar graph G whose interior faces + have roughly equal area and whose edges have roughly equal length. + + Starts from a uniform Tutte embedding (outer face on a unit circle in CCW + order starting at the south pole), then runs random-search optimization, + accepting only moves that keep the layout planar. + + Parameters + ---------- + G : networkx.Graph + A simple connected planar graph; intended for maximal planar graphs. + outer_face : tuple + Cyclic ordering of vertices on the outer face. Pinned during + optimization. + n_explore, n_refine : int + Number of iterations in the exploration and refinement phases. + explore_step, refine_step : float + Standard deviation of Gaussian moves in each phase. + w_area, w_edge : float + Weights for the face-area and edge-length penalties in the score. + seed : int + RNG seed for reproducibility. + verbose : bool + If True, print progress. + + Returns + ------- + dict[node, np.ndarray] + Position of each vertex. + """ + is_planar, emb = nx.check_planarity(G) + if not is_planar: + raise ValueError("G is not planar") + + pos = _tutte_layout(G, outer_face) + + all_faces = _get_all_faces(emb) + outer_set = set(outer_face) + interior_faces = [f for f in all_faces if set(f) != outer_set] + + if _count_crossings(G, pos) != 0: + raise RuntimeError("Initial Tutte layout is not planar; check inputs") + + interior_nodes = [v for v in G.nodes() if v not in outer_set] + if not interior_nodes: + return pos + + rng = np.random.default_rng(seed) + best_score = _score(G, pos, interior_faces, w_area, w_edge) + best_pos = {k: v.copy() for k, v in pos.items()} + initial_score = best_score + + for phase_name, n_iter, step in [("explore", n_explore, explore_step), + ("refine", n_refine, refine_step)]: + pos = {k: v.copy() for k, v in best_pos.items()} + accepted = 0 + for it in range(n_iter): + v = interior_nodes[rng.integers(len(interior_nodes))] + delta = rng.normal(0, step, 2) + old = pos[v].copy() + pos[v] = old + delta + if _count_crossings(G, pos) == 0: + new_score = _score(G, pos, interior_faces, w_area, w_edge) + if new_score < best_score: + best_score = new_score + best_pos = {k: vv.copy() for k, vv in pos.items()} + accepted += 1 + else: + pos[v] = old + else: + pos[v] = old + if verbose: + print(f" {phase_name}: score {best_score:.4f}, " + f"accepted {accepted}/{n_iter}") + + if verbose: + print(f"Initial score: {initial_score:.4f}") + print(f"Final score: {best_score:.4f}") + + return best_pos + + +# ── Demo ───────────────────────────────────────────────────────────────────── + +if __name__ == "__main__": + edges = [(0,1),(0,2),(0,3),(0,4),(0,5),(0,6), + (1,2),(1,3),(1,4),(1,5),(1,6), + (2,3),(2,4),(3,5),(4,6)] + G = nx.Graph() + G.add_nodes_from(range(7)) + G.add_edges_from(edges) + + pos = balanced_planar_layout(G, outer_face=(0, 3, 5), verbose=True) + print("\nFinal positions:") + for v in sorted(pos): + p = pos[v] + print(f" {v}: ({p[0]:+.3f}, {p[1]:+.3f})") + print(f"Crossings: {_count_crossings(G, pos)}") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage.py new file mode 100644 index 0000000..ebd497b --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage.py @@ -0,0 +1,85 @@ +""" +Coverage analysis: for each pair (n, target-class restriction), check whether +every iso-class in the restriction is reachable as a level resolution of +some triangulation on n vertices. +""" +import networkx as nx +import time +from level_cycles import all_level_resolutions +from triangulation_gen import enumerate_all_triangulations + + +def iso_class(G, reps): + for i, r in enumerate(reps): + if nx.is_isomorphic(G, r): + return i + return -1 + + +def resolution_classes(G, reps): + return {iso_class(Gp, reps) for Gp, _, _, _ in all_level_resolutions(G)} + + +def min_degree(G): + return min(d for _, d in G.degree()) + + +def coverage_report(n, target_filter=None, source_filter=None): + """ + target_filter / source_filter: callables G -> bool, or None for "any". + """ + print(f"\n{'='*60}\nn = {n}\n{'='*60}") + t0 = time.time() + reps = enumerate_all_triangulations(n) + print(f"Total iso-classes: {len(reps)}") + + if source_filter is None: + sources = list(range(len(reps))) + print(f"Sources: all {len(sources)}") + else: + sources = [i for i, G in enumerate(reps) if source_filter(G)] + print(f"Sources (filtered): {len(sources)}") + + if target_filter is None: + targets = set(range(len(reps))) + else: + targets = {i for i, G in enumerate(reps) if target_filter(G)} + print(f"Targets: {len(targets)} iso-classes") + for i in sorted(targets): + deg = sorted((d for _, d in reps[i].degree()), reverse=True) + print(f" T{i}: degree {deg}") + + reached, sources_per_target = set(), {i: [] for i in targets} + for src_i in sources: + prod = resolution_classes(reps[src_i], reps) & targets + for p in prod: + sources_per_target[p].append(src_i) + reached |= prod + + print("\nCoverage:") + for i in sorted(targets): + sources_list = sources_per_target[i] + status = "REACHED" if sources_list else "UNREACHABLE" + s = ", ".join(f"T{j}" for j in sources_list[:6]) + if len(sources_list) > 6: + s += f", ... ({len(sources_list)} total)" + elif not sources_list: + s = "(none)" + print(f" T{i}: {status} via {s}") + + uncov = targets - reached + print(f"\nUncovered: {sorted(uncov)}") + print(f"Time: {time.time()-t0:.1f}s") + + +if __name__ == "__main__": + # General coverage (any source, any target) + for n in [6, 7]: + coverage_report(n) + + # md3 sources -> md4 targets + print("\n\n" + "#"*60) + print("md3 sources -> md4 targets") + print("#"*60) + for n in [6, 7, 8]: + coverage_report(n, target_filter=lambda G: min_degree(G) >= 4) diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_chunked.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_chunked.py new file mode 100644 index 0000000..d386004 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_chunked.py @@ -0,0 +1,99 @@ +"""Chunked coverage analysis. Processes a slice of sources per invocation. +Saves and resumes progress via a pickle file. Run repeatedly until done.""" +import sys; sys.path.insert(0, '/home/claude/build') +import networkx as nx +import pickle +import time +import os +from collections import defaultdict +from level_cycles import all_level_resolutions +from triangulation_gen import enumerate_all_triangulations + +STATE_FILE = '/tmp/n11_state.pkl' +N = 11 +CHUNK_SECONDS = 200 # how long this invocation should work + +def degree_seq(G): + return tuple(sorted((d for _, d in G.degree()), reverse=True)) + +def min_degree(G): + return min(d for _, d in G.degree()) + +# Load or initialize state +if os.path.exists(STATE_FILE): + with open(STATE_FILE, 'rb') as f: + state = pickle.load(f) + print(f"Resumed. Done: {state['next_idx']}/{state['total']}") +else: + t0 = time.time() + print("Initializing...") + reps = enumerate_all_triangulations(N) + print(f" {len(reps)} iso-classes in {time.time()-t0:.1f}s") + deg_b = defaultdict(list); hash_b = defaultdict(list) + for i, G in enumerate(reps): + ds = degree_seq(G) + deg_b[ds].append(i) + hash_b[(ds, nx.weisfeiler_lehman_graph_hash(G))].append((i, G)) + state = { + 'reps': reps, + 'deg_b': dict(deg_b), + 'hash_b': dict(hash_b), + 'produced_by': {}, + 'next_idx': 0, + 'total': len(reps), + } + with open(STATE_FILE, 'wb') as f: + pickle.dump(state, f) + print(f"Init done in {time.time()-t0:.1f}s") + +reps = state['reps'] +deg_b = state['deg_b'] +hash_b = state['hash_b'] +produced_by = state['produced_by'] + +def fast_iso(G): + ds = degree_seq(G) + if ds not in deg_b: return -1 + h = nx.weisfeiler_lehman_graph_hash(G) + for idx, rep in hash_b.get((ds, h), []): + if nx.is_isomorphic(G, rep): return idx + return -1 + +start = time.time() +i = state['next_idx'] +processed = 0 +while i < state['total'] and time.time() - start < CHUNK_SECONDS: + G = reps[i] + s = set() + for Gp, _, _, _ in all_level_resolutions(G): + c = fast_iso(Gp) + if c >= 0: s.add(c) + produced_by[i] = s + i += 1 + processed += 1 + +state['next_idx'] = i +state['produced_by'] = produced_by +with open(STATE_FILE, 'wb') as f: + pickle.dump(state, f) + +elapsed = time.time() - start +print(f"Processed {processed} sources in {elapsed:.0f}s. " + f"Now at {i}/{state['total']}.") + +if i >= state['total']: + print("\n=== ALL DONE ===") + all_prod = set().union(*produced_by.values()) + uncov = set(range(state['total'])) - all_prod + md4_idx = set(j for j, G in enumerate(reps) if min_degree(G) >= 4) + md4_reached = set().union(*(produced_by[k] & md4_idx + for k in range(state['total']))) + md4_uncov = md4_idx - md4_reached + print(f"General: {state['total'] - len(uncov)}/{state['total']} reached") + if uncov: + for j in sorted(uncov)[:20]: + print(f" uncovered T{j}: degree {degree_seq(reps[j])}") + print(f"md4: {len(md4_idx) - len(md4_uncov)}/{len(md4_idx)} reached") + if md4_uncov: + for j in sorted(md4_uncov): + print(f" uncovered md4 T{j}: degree {degree_seq(reps[j])}") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_fast.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_fast.py new file mode 100644 index 0000000..bd3b8f5 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_fast.py @@ -0,0 +1,78 @@ +""" +Fast coverage analysis using WL hash for iso-class lookup. +""" +import sys; sys.path.insert(0, '/home/claude/build') +import networkx as nx +import time +from collections import defaultdict +from level_cycles import all_level_resolutions +from triangulation_gen_fast import enumerate_all_triangulations_fast, wl_hash + + +def build_iso_index(reps): + """Map WL hash -> list of (idx, G) so iso-checks only run within hash bucket.""" + idx = defaultdict(list) + for i, G in enumerate(reps): + idx[wl_hash(G)].append((i, G)) + return idx + + +def iso_class_fast(G, iso_index): + h = wl_hash(G) + for i, R in iso_index.get(h, []): + if nx.is_isomorphic(G, R): + return i + return -1 + + +def resolution_classes_fast(G, iso_index): + return {iso_class_fast(Gp, iso_index) + for Gp, _, _, _ in all_level_resolutions(G) + if iso_class_fast(Gp, iso_index) >= 0} + + +def min_degree(G): + return min(d for _, d in G.degree()) + + +def coverage_at(n, report_every=100): + t0 = time.time() + reps = enumerate_all_triangulations_fast(n) + iso_index = build_iso_index(reps) + print(f"n={n}: {len(reps)} iso-classes enumerated and indexed " + f"({time.time()-t0:.1f}s)") + md4_idx = set(i for i, G in enumerate(reps) if min_degree(G) >= 4) + + t1 = time.time() + produced_by = {} + for i, G in enumerate(reps): + produced_by[i] = resolution_classes_fast(G, iso_index) + if (i + 1) % report_every == 0: + elapsed = time.time() - t1 + eta = elapsed * (len(reps) - i - 1) / (i + 1) + print(f" {i+1}/{len(reps)} ({elapsed:.0f}s, ETA {eta:.0f}s)") + print(f"Resolution computation: {time.time()-t1:.1f}s") + + all_prod = set().union(*produced_by.values()) + uncovered = set(range(len(reps))) - all_prod + md4_reached = set().union(*(produced_by[i] & md4_idx + for i in range(len(reps)))) + md4_uncov = md4_idx - md4_reached + + print(f"\nGeneral coverage: {len(reps) - len(uncovered)}/{len(reps)}") + if uncovered: + for i in sorted(uncovered): + deg = sorted((d for _, d in reps[i].degree()), reverse=True) + print(f" uncovered T{i}: degree {deg}") + print(f"md4 coverage: {len(md4_idx) - len(md4_uncov)}/{len(md4_idx)}") + if md4_uncov: + for i in sorted(md4_uncov): + deg = sorted((d for _, d in reps[i].degree()), reverse=True) + print(f" uncovered md4 T{i}: degree {deg}") + print(f"Total: {time.time()-t0:.1f}s") + + +if __name__ == "__main__": + for n in [11]: + print("=" * 60) + coverage_at(n, report_every=100) diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_new_def.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_new_def.py new file mode 100644 index 0000000..ae7f3d7 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/coverage_new_def.py @@ -0,0 +1,104 @@ +""" +Coverage analysis under the NEW definition of level resolution: +G' is a level resolution of G via S iff both parity subgraphs of G' (using +G's BFS-from-S parities) are bipartite. + +Since the iso-class of G' doesn't depend on the specific labeling (only the +cardinalities |V_e|, |V_o| matter via permutation), G' is reachable iff: +- there exist achievable parity cardinalities (s, n-s) from some (G, S), +- G' admits a 2-partition into bipartite-induced subgraphs of those sizes. +""" +import sys; sys.path.insert(0, '/home/claude/build') +import networkx as nx +import time +from itertools import combinations +from collections import defaultdict +from level_cycles import ( + compute_levels, get_all_faces, level_sources, +) +from triangulation_gen import enumerate_all_triangulations +from triangulation_gen_fast import enumerate_all_triangulations_fast + + +def min_degree(G): + return min(d for _, d in G.degree()) + + +def achievable_parity_splits(reps): + """Return set of (|V_e|, |V_o|) cardinality tuples achievable across all + (G, S) pairs from the given triangulation reps.""" + splits = set() + for G in reps: + ip, emb = nx.check_planarity(G) + if not ip: continue + for kind, label, source_set in level_sources(G, emb): + levels = compute_levels(G, source_set) + even = sum(1 for v in levels.values() if v % 2 == 0) + odd = sum(1 for v in levels.values() if v % 2 == 1) + splits.add((even, odd)) + return splits + + +def has_bipartite_partition(G, sizes): + """Does G admit a 2-partition (S, V\\S) with |S| in `sizes` (or n-|S| in + sizes) such that both G[S] and G[V\\S] are bipartite?""" + V = list(G.nodes()) + n = len(V) + accept_sizes = set(s for s in sizes if 1 <= s <= n - 1) + accept_sizes |= set(n - s for s in sizes if 1 <= n - s <= n - 1) + seen_sizes = sorted(accept_sizes) + # Avoid duplicate work: check size s and size n-s give same partition + seen = set() + for size in seen_sizes: + if size > n - size: + continue + for S_tuple in combinations(V, size): + S = set(S_tuple) + key = frozenset(S) + if key in seen: continue + seen.add(key) + if nx.is_bipartite(G.subgraph(S)) and \ + nx.is_bipartite(G.subgraph(set(V) - S)): + return True + return False + + +def analyze(n, use_fast=False, verbose=True): + print(f"\n{'='*60}\nn = {n}\n{'='*60}") + t0 = time.time() + reps = (enumerate_all_triangulations_fast(n) if use_fast + else enumerate_all_triangulations(n)) + md4_idx = set(i for i, G in enumerate(reps) if min_degree(G) >= 4) + print(f"{len(reps)} iso-classes, {len(md4_idx)} md4") + + splits = achievable_parity_splits(reps) + achievable_sizes = set(s for s, _ in splits) + print(f"Achievable parity splits: {sorted(splits)}") + print(f"Achievable sizes: {sorted(achievable_sizes)}") + + print(f"\nReachability under NEW definition:") + reachable, unreachable_md4 = [], [] + for i, G in enumerate(reps): + if has_bipartite_partition(G, achievable_sizes): + reachable.append(i) + elif i in md4_idx: + unreachable_md4.append(i) + if verbose and i not in reachable and i in md4_idx: + deg = sorted((d for _, d in G.degree()), reverse=True) + print(f" T{i} (md4, deg {deg}): UNREACHABLE") + not_reachable = set(range(len(reps))) - set(reachable) + print(f"Reachable: {len(reachable)}/{len(reps)}") + print(f"md4 reachable: {len(md4_idx & set(reachable))}/{len(md4_idx)}") + if not_reachable: + print(f"Unreachable iso-classes: {sorted(not_reachable)}") + for i in sorted(not_reachable): + deg = sorted((d for _, d in reps[i].degree()), reverse=True) + marker = " (md4)" if i in md4_idx else "" + print(f" T{i}: degree {deg}{marker}") + print(f"Time: {time.time()-t0:.1f}s") + return reachable, unreachable_md4 + + +if __name__ == "__main__": + for n in [6, 7, 8, 9, 10, 11]: + analyze(n, use_fast=(n >= 11)) diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/face_counting.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/face_counting.py new file mode 100644 index 0000000..4c643fb --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/face_counting.py @@ -0,0 +1,125 @@ +""" +Double counting at the face/source level. +For each (G_iso, source-up-to-aut, flip choice), apply the resolution and +track which G'_iso it lands on. Then aggregate. + +For each md4 target G' we report: +- N_iso(G') = # distinct preimage iso-classes G that resolve to G' +- N_paths(G') = # distinct (G_iso, source orbit, flip choice) triples + +For each source G we report: +- f(G) = # (source, flip choice) pairs producing any md4 target +""" +import sys; sys.path.insert(0, '/home/claude/build') +import networkx as nx +from collections import defaultdict +from itertools import product as iproduct + +from triangulation_gen import enumerate_all_triangulations +from triangulation_gen_fast import enumerate_all_triangulations_fast +from level_cycles import ( + compute_levels, get_all_faces, get_level_cycles_by_parity, + get_flip_candidates, level_sources, +) + + +def min_degree(G): + return min(d for _, d in G.degree()) + + +def iso_class(G, reps): + for i, r in enumerate(reps): + if nx.is_isomorphic(G, r): + return i + return -1 + + +def all_resolutions(G): + """Yield (G_resolved, source_kind, source_label) for every resolution. + Equivalent paths are NOT deduplicated here — caller can do so via iso.""" + ip, emb = nx.check_planarity(G) + if not ip: return + for kind, label, source_set in level_sources(G, emb): + levels = compute_levels(G, source_set) + odd_cycles, even_cycles = get_level_cycles_by_parity(G, emb, levels) + cand_lists = [] + ok = True + for cycle in odd_cycles: + cands = get_flip_candidates(G, emb, cycle) + if not cands: ok = False; break + cand_lists.append(cands) + if not ok: continue + for cycle in even_cycles: + cands = get_flip_candidates(G, emb, cycle) + cand_lists.append(cands + [None]) + if not cand_lists: + yield (G.copy(), kind, label); continue + for choices in iproduct(*cand_lists): + Gp = G.copy() + applied = set(); ok2 = True + for c in choices: + if c is None: continue + u, v, w, x = c + if frozenset([u,v]) in applied: ok2 = False; break + if not Gp.has_edge(u, v) or Gp.has_edge(w, x): ok2 = False; break + Gp.remove_edge(u, v); Gp.add_edge(w, x) + applied.add(frozenset([u, v])) + if not ok2: continue + ipp, _ = nx.check_planarity(Gp) + if not ipp: continue + yield (Gp, kind, label) + + +def analyze(n, use_fast=False): + print(f"\n{'='*70}\nn = {n}\n{'='*70}") + if use_fast: + reps = enumerate_all_triangulations_fast(n) + else: + reps = enumerate_all_triangulations(n) + md4_idx = [i for i, G in enumerate(reps) if min_degree(G) >= 4] + md4_set = set(md4_idx) + + # For each source iso-class, compute # paths to each target iso-class + pre_count = defaultdict(lambda: defaultdict(int)) # target -> source -> count + src_paths_md4 = defaultdict(int) # source -> total paths landing on md4 + + print(f"Computing resolutions for {len(reps)} sources...", flush=True) + for src_i, G in enumerate(reps): + # Compute all distinct (target iso, source orbit) pairs from this G + for Gp, kind, label in all_resolutions(G): + tgt = iso_class(Gp, reps) + if tgt < 0: continue + pre_count[tgt][src_i] += 1 + if tgt in md4_set: + src_paths_md4[src_i] += 1 + + # Report by md4 target + print(f"\n{'target':28s} {'N_iso':>6} {'N_paths':>9} {'min path/iso':>14}") + print("-" * 70) + for tgt in md4_idx: + deg = sorted((d for _, d in reps[tgt].degree()), reverse=True) + sources_hitting = list(pre_count[tgt].keys()) + N_iso = len(sources_hitting) + N_paths = sum(pre_count[tgt].values()) + min_paths_per_src = (min(pre_count[tgt].values()) + if sources_hitting else 0) + print(f"{str(deg):28s} {N_iso:>6} {N_paths:>9} {min_paths_per_src:>14}") + # Double-counting check: + total_md4_paths = sum(sum(d.values()) for d in pre_count.values() + if any(k in md4_set for k in [...])) + # Simpler total + total_paths = sum(src_paths_md4.values()) + avg_per_target = total_paths / len(md4_idx) if md4_idx else 0 + print("-" * 70) + print(f"Total md4 paths: {total_paths}") + print(f"Average per md4 target: {avg_per_target:.1f}") + if md4_idx: + min_N_iso = min(len(pre_count[t]) for t in md4_idx) + print(f"Minimum N_iso over md4 targets: {min_N_iso}") + + return pre_count, md4_idx + + +if __name__ == "__main__": + for n in [6, 7, 8, 9, 10]: + analyze(n, use_fast=(n >= 10)) diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/four_color.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/four_color.py new file mode 100644 index 0000000..d4b93d3 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/four_color.py @@ -0,0 +1,98 @@ +""" +4-color G' using the level structure from G: + - Even-level subgraph: 2-color with RED/BLUE via BFS + - Odd-level subgraph: 2-color with YELLOW/GREEN via BFS + +This succeeds iff each parity subgraph is bipartite — which is the goal of +the level resolution. If a parity subgraph contains an odd cycle, BFS will +find a conflict and we report which edge violates the 2-coloring. +""" +import networkx as nx +import numpy as np +from collections import deque + + +def two_color_subgraph(G_sub, color_a, color_b): + """ + Two-color the (possibly disconnected) subgraph via BFS, alternating + color_a / color_b by BFS distance parity from a root in each component. + + Returns: + coloring: dict[node, color] + bad_edges: list of edges where adjacent vertices got the same colour + (empty iff subgraph is bipartite) + """ + coloring = {} + bad_edges = [] + for start in G_sub.nodes(): + if start in coloring: + continue + coloring[start] = color_a + queue = deque([start]) + while queue: + v = queue.popleft() + for w in G_sub.neighbors(v): + if w not in coloring: + coloring[w] = color_b if coloring[v] == color_a else color_a + queue.append(w) + elif coloring[w] == coloring[v]: + e = tuple(sorted([v, w])) + if e not in bad_edges: + bad_edges.append(e) + return coloring, bad_edges + + +def four_color_via_levels(G_prime, levels): + """ + 4-color G' using level labels from G. + Even-level vertices get RED/BLUE; odd-level get YELLOW/GREEN. + + Returns: + coloring: dict[node, str] + bad_edges: dict with keys 'even', 'odd', 'cross' for violations + within each parity subgraph and between them + (the cross list should always be empty by construction) + """ + even_nodes = [v for v in G_prime.nodes() if levels[v] % 2 == 0] + odd_nodes = [v for v in G_prime.nodes() if levels[v] % 2 == 1] + + even_sub = G_prime.subgraph(even_nodes).copy() + odd_sub = G_prime.subgraph(odd_nodes).copy() + + coloring_even, bad_even = two_color_subgraph(even_sub, 'red', 'blue') + coloring_odd, bad_odd = two_color_subgraph(odd_sub, 'yellow', 'green') + + coloring = {**coloring_even, **coloring_odd} + + cross_bad = [] + for u, v in G_prime.edges(): + if coloring[u] == coloring[v]: + cross_bad.append(tuple(sorted([u, v]))) + + return coloring, { + 'even': bad_even, + 'odd': bad_odd, + 'cross': [e for e in cross_bad + if tuple(sorted(e)) not in bad_even + and tuple(sorted(e)) not in bad_odd], + } + + +if __name__ == "__main__": + # Quick demo on the n=7 graph used earlier + edges = [(0,1),(0,2),(0,3),(0,4),(0,5),(0,6), + (1,2),(1,3),(1,4),(1,5),(1,6), + (2,3),(2,4),(3,5),(4,6)] + G = nx.Graph(); G.add_nodes_from(range(7)); G.add_edges_from(edges) + Gp = G.copy(); Gp.remove_edge(1, 2); Gp.add_edge(4, 3) + + levels = {0:0, 3:0, 5:0, 1:1, 2:1, 4:1, 6:1} + coloring, bad = four_color_via_levels(Gp, levels) + print("Coloring:") + for v in sorted(coloring): + print(f" vertex {v} (level {levels[v]}): {coloring[v]}") + print(f"\nBad edges in even subgraph: {bad['even']}") + print(f"Bad edges in odd subgraph: {bad['odd']}") + print(f"Cross-parity bad edges: {bad['cross']}") + print(f"\nValid 4-coloring: " + f"{not any(bad.values())}") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/four_color_viz.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/four_color_viz.py new file mode 100644 index 0000000..e6747ee --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/four_color_viz.py @@ -0,0 +1,256 @@ +""" +For each n=7 example (G → G' covering all 5 iso-classes), apply the level +4-coloring to G' using levels from G, and visualize. +""" +import sys; sys.path.insert(0, '/home/claude') +import networkx as nx +import matplotlib.pyplot as plt +import matplotlib.patches as mpatches +import numpy as np +from collections import deque +from itertools import combinations, product as iproduct + +from balanced_layout import balanced_planar_layout, _get_all_faces +from four_color import four_color_via_levels + +def compute_levels(G, outer_face): + levels, queue = {}, deque() + for v in outer_face: + levels[v] = 0; queue.append(v) + while queue: + v = queue.popleft() + for w in G.neighbors(v): + if w not in levels: + levels[w] = levels[v] + 1; queue.append(w) + return levels + +def get_odd_level_cycles(emb, levels): + odd = [] + for face in _get_all_faces(emb): + lv = {levels.get(v) for v in face} + if len(lv) == 1 and None not in lv and len(face) % 2 == 1: + odd.append(face) + return odd + +def get_flip_candidates(G, emb, cycle): + cands = [] + n = len(cycle) + for i in range(n): + u, v = cycle[i], cycle[(i+1) % n] + if not G.has_edge(u, v): continue + f1 = emb.traverse_face(u, v); f2 = emb.traverse_face(v, u) + if len(f1) != 3 or len(f2) != 3: continue + w = next(x for x in f1 if x != u and x != v) + x = next(y for y in f2 if y != u and y != v) + if w == x or G.has_edge(w, x): continue + cands.append((u, v, w, x)) + return cands + +def is_triangulation(G): + n = G.number_of_nodes() + if G.number_of_edges() != 3*n - 6: return False + if not nx.is_connected(G): return False + ip, emb = nx.check_planarity(G) + if not ip: return False + seen = set() + for v in emb.nodes(): + for w in emb[v]: + if (v, w) not in seen: + f = emb.traverse_face(v, w) + for i in range(len(f)): + seen.add((f[i], f[(i+1) % len(f)])) + if len(f) != 3: return False + return True + +# ── Reps ───────────────────────────────────────────────────────────────────── +nodes = list(range(7)) +all_edges = list(combinations(nodes, 2)) +reps = [] +for sub in combinations(all_edges, 15): + G = nx.Graph(); G.add_nodes_from(nodes); G.add_edges_from(sub) + if is_triangulation(G): + if all(not nx.is_isomorphic(G, r) for r in reps): + reps.append(G) + if len(reps) == 5: break + +def iso_class(G): + for i, r in enumerate(reps): + if nx.is_isomorphic(G, r): return i + return -1 + +# Same pairs as before +desired_pairs = [(0, 2), (1, 4), (2, 0), (3, 1), (4, 3)] + +def find_example(src_idx, tgt_idx): + G = reps[src_idx] + ip, emb = nx.check_planarity(G) + for outer_face in _get_all_faces(emb): + levels = compute_levels(G, outer_face) + odd_cycles = get_odd_level_cycles(emb, levels) + if not odd_cycles: continue + candidate_lists = [] + ok = True + for cycle in odd_cycles: + cands = get_flip_candidates(G, emb, cycle) + if not cands: ok = False; break + candidate_lists.append(cands) + if not ok: continue + for choices in iproduct(*candidate_lists): + Gp = G.copy() + valid = True + for u, v, w, x in choices: + if not Gp.has_edge(u, v) or Gp.has_edge(w, x): + valid = False; break + Gp.remove_edge(u, v); Gp.add_edge(w, x) + if valid and is_triangulation(Gp) and iso_class(Gp) == tgt_idx: + return G, outer_face, choices, Gp + return None + +examples = [find_example(s, t) for s, t in desired_pairs] +print("Examples found:", sum(1 for e in examples if e is not None)) + +# ── 4-color and visualize ──────────────────────────────────────────────────── + +color_hex = {'red':'#e53935','blue':'#1e88e5', + 'yellow':'#fdd835','green':'#43a047'} + +fig, axes = plt.subplots(len(examples), 2, figsize=(14, 5.5 * len(examples))) +if len(examples) == 1: + axes = np.array([axes]) + +print("\nColoring each G':") +for row, ex in enumerate(examples): + if ex is None: continue + G, outer_face, choices, Gp = ex + src, tgt = desired_pairs[row] + + levels_G = compute_levels(G, outer_face) + coloring, bad = four_color_via_levels(Gp, levels_G) + valid = not any(bad.values()) + print(f" T{src}→T{tgt}: valid 4-coloring = {valid}") + if not valid: + print(f" bad even: {bad['even']}, bad odd: {bad['odd']}") + + # Layout for G and G' + _, emb_G = nx.check_planarity(G) + _, emb_p = nx.check_planarity(Gp) + Gp_faces = _get_all_faces(emb_p) + outer_set = set(outer_face) + Gp_outer = max(Gp_faces, + key=lambda f: (len(set(f) & outer_set), -len(f))) + pos_G = balanced_planar_layout(G, outer_face, n_explore=4000, + n_refine=2000, seed=42 + src) + pos_Gp = balanced_planar_layout(Gp, Gp_outer, n_explore=4000, + n_refine=2000, seed=42 + tgt) + + # ── Draw G on the left (no 4-coloring; just structure) ──────────────── + ax_G = axes[row, 0] + odd_cycles_G = get_odd_level_cycles(emb_G, levels_G) + ax_G.add_patch(plt.Polygon([pos_G[v] for v in outer_face], closed=True, + facecolor='#E3F2FD', alpha=0.5, + edgecolor='none', zorder=0)) + for cycle in odd_cycles_G: + if set(cycle) == outer_set: continue + ax_G.add_patch(plt.Polygon([pos_G[v] for v in cycle], closed=True, + facecolor='#FFF9C4', alpha=0.6, + edgecolor='none', zorder=0)) + flip_edges_G = set(frozenset([u, v]) for u, v, _, _ in choices) + for u, v in G.edges(): + if frozenset([u, v]) in flip_edges_G: + nx.draw_networkx_edges(G, pos_G, edgelist=[(u, v)], ax=ax_G, + edge_color='#e53935', width=3.5, + style='dashed') + elif levels_G[u] == levels_G[v]: + nx.draw_networkx_edges(G, pos_G, edgelist=[(u, v)], ax=ax_G, + edge_color='#2196F3', width=2.5) + else: + nx.draw_networkx_edges(G, pos_G, edgelist=[(u, v)], ax=ax_G, + edge_color='#bdbdbd', width=1.5) + # Nodes by level (un-coloured) + level_palette = ['#1565C0', '#E65100', '#2E7D32'] + node_lv_cols = [level_palette[min(levels_G[v], len(level_palette)-1)] + for v in G.nodes()] + nx.draw_networkx_nodes(G, pos_G, ax=ax_G, node_color=node_lv_cols, + node_size=550) + nx.draw_networkx_labels(G, pos_G, ax=ax_G, font_size=11, + font_color='white', font_weight='bold') + for v, p in pos_G.items(): + ax_G.annotate(f'L{levels_G[v]}', xy=p + np.array([0.07, 0.07]), + fontsize=8, color='#333', zorder=5) + flip_str = ", ".join(f"({u},{v})→({w},{x})" for u, v, w, x in choices) + ax_G.set_title(f"G (T{src}) | outer face {outer_face}\n" + f"flips: {flip_str}", + fontsize=9, fontweight='bold', pad=8) + ax_G.set_xlim(-1.5, 1.5); ax_G.set_ylim(-1.5, 1.5) + ax_G.set_aspect('equal'); ax_G.axis('off') + + # ── Draw G' on the right (4-coloured) ───────────────────────────────── + ax = axes[row, 1] + ax.add_patch(plt.Polygon([pos_Gp[v] for v in Gp_outer], closed=True, + facecolor='#ECEFF1', alpha=0.6, + edgecolor='none', zorder=0)) + + bad_edge_set = set() + for e in bad['even'] + bad['odd'] + bad['cross']: + bad_edge_set.add(frozenset(e)) + + for u, v in Gp.edges(): + key = frozenset([u, v]) + same_level = levels_G[u] == levels_G[v] + if key in bad_edge_set: + nx.draw_networkx_edges(Gp, pos_Gp, edgelist=[(u, v)], ax=ax, + edge_color='#000', width=4.0) + elif same_level: + nx.draw_networkx_edges(Gp, pos_Gp, edgelist=[(u, v)], ax=ax, + edge_color='#666', width=2.5) + else: + nx.draw_networkx_edges(Gp, pos_Gp, edgelist=[(u, v)], ax=ax, + edge_color='#bdbdbd', width=1.5) + + node_colors_4 = [color_hex[coloring[v]] for v in Gp.nodes()] + nx.draw_networkx_nodes(Gp, pos_Gp, ax=ax, + node_color=node_colors_4, + edgecolors='black', linewidths=2, + node_size=600) + nx.draw_networkx_labels(Gp, pos_Gp, ax=ax, font_size=11, + font_color='white', font_weight='bold') + for v, p in pos_Gp.items(): + ax.annotate(f'L{levels_G[v]}', xy=p + np.array([0.07, 0.07]), + fontsize=8, color='#333', zorder=5) + + title = (f"G' (T{tgt}) | layout outer {Gp_outer}\n" + f"valid 4-coloring: {valid}") + if not valid: + title += f"\nconflicts: even={bad['even']} odd={bad['odd']}" + ax.set_title(title, fontsize=9, fontweight='bold', pad=8) + ax.set_xlim(-1.5, 1.5); ax.set_ylim(-1.5, 1.5) + ax.set_aspect('equal'); ax.axis('off') + +legend_elements = [ + mpatches.Patch(facecolor='#E3F2FD', edgecolor='none', + label="G outer face"), + mpatches.Patch(facecolor='#FFF9C4', edgecolor='none', + label='G odd level cycle'), + plt.Line2D([0],[0], color='#e53935', lw=3.0, ls='dashed', + label='Edge flipped out (in G)'), + plt.Line2D([0],[0], color='#2196F3', lw=2.5, label='Level edge'), + mpatches.Patch(facecolor=color_hex['red'], edgecolor='black', + label="G' red (even-level)"), + mpatches.Patch(facecolor=color_hex['blue'], edgecolor='black', + label="G' blue (even-level)"), + mpatches.Patch(facecolor=color_hex['yellow'], edgecolor='black', + label="G' yellow (odd-level)"), + mpatches.Patch(facecolor=color_hex['green'], edgecolor='black', + label="G' green (odd-level)"), + plt.Line2D([0],[0], color='#000', lw=4.0, + label='Coloring conflict edge'), +] +fig.legend(handles=legend_elements, loc='lower center', ncol=4, + fontsize=10, bbox_to_anchor=(0.5, -0.002)) +plt.suptitle("4-coloring of G' using levels of G " + "(even=red/blue, odd=yellow/green)", + fontsize=13, fontweight='bold', y=1.0) +plt.tight_layout() +plt.savefig('/mnt/user-data/outputs/four_color.png', + dpi=130, bbox_inches='tight') +print("\nSaved.") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/level_cycles.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/level_cycles.py new file mode 100644 index 0000000..5939fc0 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/level_cycles.py @@ -0,0 +1,173 @@ +""" +Level cycle computation for maximal planar graphs. + +Definitions +----------- +- Level source: either (a) a face of G [all face vertices at level 0] + or (b) a degree-3 vertex of G [singleton at level 0] +- Levels: BFS distance from the level source. +- Level cycle: simple face of the level subgraph L_k (subgraph induced by + level-k vertices) in the embedding inherited from G's planar embedding. +- Level resolution: G' obtained from G by flipping + - exactly one edge per ODD level cycle (mandatory) + - at most one edge per EVEN level cycle (optional) +""" +import networkx as nx +from collections import deque, defaultdict +from itertools import product as iproduct + + +def compute_levels(G, source_set): + """BFS levels from any iterable of source vertices.""" + levels, queue = {}, deque() + for v in source_set: + levels[v] = 0 + queue.append(v) + while queue: + v = queue.popleft() + for w in G.neighbors(v): + if w not in levels: + levels[w] = levels[v] + 1 + queue.append(w) + return levels + + +def get_all_faces(emb): + seen, faces = set(), [] + for v in emb.nodes(): + for w in emb[v]: + if (v, w) not in seen: + face = emb.traverse_face(v, w) + for i in range(len(face)): + seen.add((face[i], face[(i+1) % len(face)])) + faces.append(tuple(face)) + return faces + + +def inherited_embedding(emb_G, sub_nodes): + """PlanarEmbedding of the induced subgraph with inherited cyclic order.""" + sub_set = set(sub_nodes) + emb = nx.PlanarEmbedding() + for v in sub_nodes: + emb.add_node(v) + for v in sub_nodes: + cw = [w for w in emb_G.neighbors_cw_order(v) if w in sub_set] + prev = None + for w in cw: + if prev is None: + emb.add_half_edge_first(v, w) + else: + emb.add_half_edge_cw(v, w, prev) + prev = w + return emb + + +def _is_simple(face): + return len(set(face)) == len(face) + + +def get_level_cycles_by_parity(G, emb_G, levels): + """Simple faces of each L_k, split by length parity.""" + by_level = defaultdict(list) + for v, lv in levels.items(): + by_level[lv].append(v) + odd, even, seen = [], [], set() + for k, nodes_k in by_level.items(): + if len(nodes_k) < 3: + continue + L_k = G.subgraph(nodes_k) + if L_k.number_of_edges() < 3: + continue + try: + emb_L = inherited_embedding(emb_G, nodes_k) + faces = get_all_faces(emb_L) + except Exception: + ip, emb_L = nx.check_planarity(L_k) + if not ip: + continue + faces = get_all_faces(emb_L) + for face in faces: + if not _is_simple(face): + continue + rots = [tuple(face[i:] + face[:i]) for i in range(len(face))] \ + + [tuple(face[::-1][i:] + face[::-1][:i]) + for i in range(len(face))] + canon = min(rots) + if canon in seen: + continue + seen.add(canon) + (odd if len(face) % 2 else even).append(face) + return odd, even + + +def get_flip_candidates(G, emb_G, cycle): + """Valid edge flips along a level cycle.""" + cands = [] + n = len(cycle) + for i in range(n): + u, v = cycle[i], cycle[(i+1) % n] + if not G.has_edge(u, v): + continue + f1 = emb_G.traverse_face(u, v) + f2 = emb_G.traverse_face(v, u) + if len(f1) != 3 or len(f2) != 3: + continue + w = next(x for x in f1 if x != u and x != v) + x = next(y for y in f2 if y != u and y != v) + if w == x or G.has_edge(w, x): + continue + cands.append((u, v, w, x)) + return cands + + +def level_sources(G, emb_G): + """Yield every valid level source as (kind, label, vertex_set).""" + for face in get_all_faces(emb_G): + yield ('face', tuple(face), set(face)) + for v in G.nodes(): + if G.degree(v) == 3: + yield ('vertex', v, {v}) + + +def all_level_resolutions(G): + """Yield (G', source_kind, source_label, choices) for every resolution.""" + is_planar, emb = nx.check_planarity(G) + if not is_planar: + return + for kind, label, source_set in level_sources(G, emb): + levels = compute_levels(G, source_set) + odd_cycles, even_cycles = get_level_cycles_by_parity(G, emb, levels) + if not odd_cycles and not even_cycles: + yield (G.copy(), kind, label, []) + continue + cand_lists, ok = [], True + for cycle in odd_cycles: + cands = get_flip_candidates(G, emb, cycle) + if not cands: + ok = False + break + cand_lists.append(cands) + if not ok: + continue + for cycle in even_cycles: + cands = get_flip_candidates(G, emb, cycle) + cand_lists.append(cands + [None]) + for choices in iproduct(*cand_lists): + Gp = G.copy() + applied, ok2 = set(), True + for choice in choices: + if choice is None: + continue + u, v, w, x = choice + if frozenset([u, v]) in applied: + ok2 = False; break + if not Gp.has_edge(u, v) or Gp.has_edge(w, x): + ok2 = False; break + Gp.remove_edge(u, v) + Gp.add_edge(w, x) + applied.add(frozenset([u, v])) + if not ok2: + continue + ip2, _ = nx.check_planarity(Gp) + if ip2: + yield (Gp, kind, label, [c for c in choices if c is not None]) diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/n11_chunk.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/n11_chunk.py new file mode 100644 index 0000000..c4173cb --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/n11_chunk.py @@ -0,0 +1,57 @@ +""" +Process a chunk of source indices at n=11. Save accumulating results to JSON. +Usage: python3 n11_chunk.py +""" +import sys, os, json, time +sys.path.insert(0, '/home/claude/build') +from coverage_fast import ( + enumerate_all_triangulations_fast, build_iso_index, + resolution_classes_fast, min_degree, +) +import pickle + +STATE_PKL = '/tmp/n11_state.pkl' +RESULTS_JSON = '/tmp/n11_results.json' + +# Load or build state +if os.path.exists(STATE_PKL): + with open(STATE_PKL, 'rb') as f: + state = pickle.load(f) + reps = state['reps']; iso_index = state['iso_index'] + print(f"Loaded {len(reps)} reps from cache") +else: + t0 = time.time() + reps = enumerate_all_triangulations_fast(11) + iso_index = build_iso_index(reps) + print(f"Built {len(reps)} reps in {time.time()-t0:.1f}s") + with open(STATE_PKL, 'wb') as f: + pickle.dump({'reps': reps, 'iso_index': iso_index}, f) + +# Load partial results +if os.path.exists(RESULTS_JSON): + with open(RESULTS_JSON) as f: + results = {int(k): v for k, v in json.load(f).items()} +else: + results = {} + +start = int(sys.argv[1]); end = int(sys.argv[2]) +end = min(end, len(reps)) +t1 = time.time() +for i in range(start, end): + if i in results: + continue + res = resolution_classes_fast(reps[i], iso_index) + results[i] = sorted(res) + if (i + 1) % 25 == 0: + elapsed = time.time() - t1 + done = i + 1 - start + eta = elapsed * (end - start - done) / max(done, 1) + print(f" {i+1}/{end} ({elapsed:.0f}s, ETA {eta:.0f}s)", flush=True) + if (i + 1) % 50 == 0: + with open(RESULTS_JSON, 'w') as f: + json.dump({str(k): v for k, v in results.items()}, f) + +with open(RESULTS_JSON, 'w') as f: + json.dump({str(k): v for k, v in results.items()}, f) +print(f"Saved {len(results)} source results. Range [{start},{end}). " + f"Chunk time: {time.time()-t1:.1f}s") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/n7_examples.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/n7_examples.py new file mode 100644 index 0000000..165a8d4 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/n7_examples.py @@ -0,0 +1,251 @@ +""" +Generate G → G' examples covering all 5 iso-classes on n=7, +each as source and as target. Uses balanced planar layout. +""" +import sys +sys.path.insert(0, '/home/claude') + +import networkx as nx +import matplotlib.pyplot as plt +import matplotlib.patches as mpatches +import numpy as np +from collections import deque +from itertools import combinations, product as iproduct + +from balanced_layout import balanced_planar_layout, _get_all_faces + +# ── Helpers ────────────────────────────────────────────────────────────────── + +def compute_levels(G, outer_face): + levels, queue = {}, deque() + for v in outer_face: + levels[v] = 0; queue.append(v) + while queue: + v = queue.popleft() + for w in G.neighbors(v): + if w not in levels: + levels[w] = levels[v] + 1; queue.append(w) + return levels + +def get_odd_level_cycles(emb, levels): + odd = [] + for face in _get_all_faces(emb): + lv = {levels.get(v) for v in face} + if len(lv) == 1 and None not in lv and len(face) % 2 == 1: + odd.append(face) + return odd + +def get_flip_candidates(G, emb, cycle): + cands = [] + n = len(cycle) + for i in range(n): + u, v = cycle[i], cycle[(i+1) % n] + if not G.has_edge(u, v): continue + f1 = emb.traverse_face(u, v); f2 = emb.traverse_face(v, u) + if len(f1) != 3 or len(f2) != 3: continue + w = next(x for x in f1 if x != u and x != v) + x = next(y for y in f2 if y != u and y != v) + if w == x or G.has_edge(w, x): continue + cands.append((u, v, w, x)) + return cands + +def is_triangulation(G): + n = G.number_of_nodes() + if G.number_of_edges() != 3*n - 6: return False + if not nx.is_connected(G): return False + ip, emb = nx.check_planarity(G) + if not ip: return False + seen = set() + for v in emb.nodes(): + for w in emb[v]: + if (v, w) not in seen: + f = emb.traverse_face(v, w) + for i in range(len(f)): + seen.add((f[i], f[(i+1) % len(f)])) + if len(f) != 3: return False + return True + +# ── Enumerate iso-class representatives for n=7 ────────────────────────────── + +print("Enumerating triangulations on 7 vertices...") +nodes = list(range(7)) +all_edges = list(combinations(nodes, 2)) +reps = [] +for sub in combinations(all_edges, 15): + G = nx.Graph(); G.add_nodes_from(nodes); G.add_edges_from(sub) + if is_triangulation(G): + if all(not nx.is_isomorphic(G, r) for r in reps): + reps.append(G) + if len(reps) == 5: break +print(f"Found {len(reps)} reps.") +for i, G in enumerate(reps): + deg_seq = sorted((d for _, d in G.degree()), reverse=True) + print(f" T{i}: degree {deg_seq}") + +def iso_class(G): + for i, r in enumerate(reps): + if nx.is_isomorphic(G, r): return i + return -1 + +# ── Find one (G_source, outer_face, flip, G_target) for each desired pair ─── + +desired_pairs = [ + (0, 2), # T0 -> T2 + (1, 4), # T1 -> T4 + (2, 0), # T2 -> T0 + (3, 1), # T3 -> T1 + (4, 3), # T4 -> T3 +] + +def find_example(src_idx, tgt_idx): + G = reps[src_idx] + ip, emb = nx.check_planarity(G) + for outer_face in _get_all_faces(emb): + levels = compute_levels(G, outer_face) + odd_cycles = get_odd_level_cycles(emb, levels) + if not odd_cycles: continue + candidate_lists = [] + ok = True + for cycle in odd_cycles: + cands = get_flip_candidates(G, emb, cycle) + if not cands: ok = False; break + candidate_lists.append(cands) + if not ok: continue + for choices in iproduct(*candidate_lists): + Gp = G.copy() + valid = True + for u, v, w, x in choices: + if not Gp.has_edge(u, v) or Gp.has_edge(w, x): + valid = False; break + Gp.remove_edge(u, v); Gp.add_edge(w, x) + if valid and is_triangulation(Gp) and iso_class(Gp) == tgt_idx: + return G, outer_face, choices, Gp + return None + +print("\nFinding example flips for each pair:") +examples = [] +for src, tgt in desired_pairs: + ex = find_example(src, tgt) + if ex is None: + print(f" T{src} -> T{tgt}: NO EXAMPLE FOUND") + continue + G, outer_face, choices, Gp = ex + print(f" T{src} -> T{tgt}: outer={outer_face}, flips={list(choices)}") + examples.append((src, tgt, G, outer_face, choices, Gp)) + +# ── Layout and plot ────────────────────────────────────────────────────────── + +print("\nComputing balanced layouts (this takes a minute)...") +fig, axes = plt.subplots(len(examples), 2, figsize=(13, 5.5 * len(examples))) +if len(examples) == 1: + axes = np.array([axes]) + +palette = ['#1565C0', '#E65100', '#2E7D32'] + +for row, (src, tgt, G, outer_face, choices, Gp) in enumerate(examples): + levels = compute_levels(G, outer_face) + _, emb = nx.check_planarity(G) + _, emb_p = nx.check_planarity(Gp) + odd_cycles_G = get_odd_level_cycles(emb, levels) + # Re-compute levels for Gp using the same outer face (same level-0 vertices) + levels_p = compute_levels(Gp, outer_face) + odd_cycles_Gp = get_odd_level_cycles(emb_p, levels_p) + + pos_G = balanced_planar_layout(G, outer_face, n_explore=4000, + n_refine=2000, seed=42 + src) + + # G' may have a different planar embedding; the original outer face + # might no longer be a face. Pick a face of G' that maximally overlaps + # the original outer face vertices. + Gp_faces = _get_all_faces(emb_p) + outer_set_orig = set(outer_face) + Gp_outer = max(Gp_faces, + key=lambda f: (len(set(f) & outer_set_orig), -len(f))) + pos_Gp = balanced_planar_layout(Gp, Gp_outer, n_explore=4000, + n_refine=2000, seed=42 + tgt) + print(f" T{src}→T{tgt}: layouts done") + + flip_edges = set(frozenset([u, v]) for u, v, _, _ in choices) + new_edges = set(frozenset([w, x]) for _, _, w, x in choices) + + def classify_G(e): + if frozenset(e) in flip_edges: return 'flip' + eu, ev = e + if levels.get(eu) == levels.get(ev): return 'level' + return 'normal' + + def classify_Gp(e): + if frozenset(e) in new_edges: return 'new' + eu, ev = e + if levels_p.get(eu) == levels_p.get(ev): return 'level' + return 'normal' + + outer_set = set(outer_face) + + def draw(ax, Gd, pos, levels_d, odd_cycles_d, classifier, of, title): + # Outer face + ax.add_patch(plt.Polygon([pos[nd] for nd in of], + closed=True, facecolor='#E3F2FD', + alpha=0.5, edgecolor='none', zorder=0)) + # Interior odd cycles + of_set = set(of) + for cycle in odd_cycles_d: + if set(cycle) == of_set: continue + ax.add_patch(plt.Polygon([pos[nd] for nd in cycle], + closed=True, facecolor='#FFF9C4', + alpha=0.6, edgecolor='none', zorder=0)) + # Edges + cats = {'normal': [], 'level': [], 'flip': [], 'new': []} + for e in Gd.edges(): + cats[classifier(e)].append(e) + style = { + 'normal': dict(edge_color='#bdbdbd', width=1.5), + 'level': dict(edge_color='#2196F3', width=2.5), + 'flip': dict(edge_color='#e53935', width=3.5, style='dashed'), + 'new': dict(edge_color='#43a047', width=3.5), + } + for cat, eds in cats.items(): + if eds: + nx.draw_networkx_edges(Gd, pos, edgelist=eds, ax=ax, + **style[cat]) + # Nodes coloured by level + node_colors = [palette[min(levels_d.get(nd, 0), len(palette)-1)] + for nd in Gd.nodes()] + nx.draw_networkx_nodes(Gd, pos, ax=ax, node_color=node_colors, + node_size=450) + nx.draw_networkx_labels(Gd, pos, ax=ax, font_size=11, + font_color='white', font_weight='bold') + for nd, p in pos.items(): + lv = levels_d.get(nd, '?') + ax.annotate(f'L{lv}', xy=p + np.array([0.06, 0.06]), + fontsize=7, color='#444', zorder=5) + ax.set_title(title, fontsize=10, fontweight='bold', pad=8) + ax.set_xlim(-1.5, 1.5); ax.set_ylim(-1.5, 1.5) + ax.set_aspect('equal'); ax.axis('off') + + flip_str = ", ".join(f"({u},{v})→({w},{x})" for u,v,w,x in choices) + draw(axes[row, 0], G, pos_G, levels, odd_cycles_G, classify_G, + outer_face, + f"G (T{src}) | outer face {outer_face}\nflip: {flip_str}") + draw(axes[row, 1], Gp, pos_Gp, levels_p, odd_cycles_Gp, classify_Gp, + Gp_outer, + f"G' (T{tgt}) | outer face {Gp_outer}") + +legend_elements = [ + mpatches.Patch(facecolor='#E3F2FD', edgecolor='none', label='Outer face'), + mpatches.Patch(facecolor='#FFF9C4', edgecolor='none', label='Odd level cycle'), + plt.Line2D([0],[0], color='#bdbdbd', lw=1.5, label='Normal edge'), + plt.Line2D([0],[0], color='#2196F3', lw=2.5, label='Level edge'), + plt.Line2D([0],[0], color='#e53935', lw=3.0, ls='dashed', + label='Removed edge'), + plt.Line2D([0],[0], color='#43a047', lw=3.0, label='Added edge'), +] +fig.legend(handles=legend_elements, loc='lower center', ncol=6, + fontsize=10, bbox_to_anchor=(0.5, -0.005)) +plt.suptitle( + "Level resolutions on n=7 | every iso-class appears as source and target", + fontsize=14, fontweight='bold', y=1.0) +plt.tight_layout() +plt.savefig('/mnt/user-data/outputs/n7_all_classes.png', + dpi=130, bbox_inches='tight') +print("\nSaved.") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/orbit_check.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/orbit_check.py new file mode 100644 index 0000000..7c7543e --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/orbit_check.py @@ -0,0 +1,123 @@ +""" +Extended orbit-counting framework: REVERSE-FLIPS ON ANY SET OF EDGES. + +For each md4 target G' and each k ∈ {1, 2, ...}: +- Enumerate distinct (up to iso) k-flip reverse-preimages G of G' +- For each, check if G level-resolves to G' via any source + +Surjectivity at G' = some k and some preimage works. + +Includes a critical test: the icosahedron at n=12 (no deg-4 vertices). +""" +import sys; sys.path.insert(0, '/home/claude/build') +import networkx as nx +import time +from collections import defaultdict +from itertools import combinations, product as iproduct + +from level_cycles import ( + compute_levels, get_all_faces, get_level_cycles_by_parity, + get_flip_candidates, level_sources, +) + + +def reverse_flip_edge(G_prime, e): + u, v = tuple(e) + if not G_prime.has_edge(u, v): return None + _, emb = nx.check_planarity(G_prime) + f1 = emb.traverse_face(u, v); f2 = emb.traverse_face(v, u) + if len(f1) != 3 or len(f2) != 3: return None + a = next(x for x in f1 if x != u and x != v) + b = next(x for x in f2 if x != u and x != v) + if a == b or G_prime.has_edge(a, b): return None + G = G_prime.copy() + G.remove_edge(u, v); G.add_edge(a, b) + ip, _ = nx.check_planarity(G) + if not ip: return None + return G + + +def k_flip_preimages_iso(G_prime, k): + """Yield distinct (up to iso) triangulations obtainable from G_prime by + k successive reverse-flips.""" + seen = [G_prime] + frontier = [G_prime] + for step in range(k): + new_frontier = [] + new_seen = [] + for H in frontier: + for e in list(H.edges()): + Hp = reverse_flip_edge(H, frozenset(e)) + if Hp is None: continue + if any(nx.is_isomorphic(Hp, S) for S in seen + new_seen): + continue + new_seen.append(Hp) + new_frontier.append(Hp) + seen.extend(new_seen) + frontier = new_frontier + if not frontier: break + return frontier # candidates at depth EXACTLY k + + +def resolves_to_target(G, target): + """Does G level-resolve to iso(target) via any source?""" + ip, emb = nx.check_planarity(G) + if not ip: return False + for kind, label, source_set in level_sources(G, emb): + levels = compute_levels(G, source_set) + odd_cycles, even_cycles = get_level_cycles_by_parity(G, emb, levels) + cand_lists, ok = [], True + for cycle in odd_cycles: + cands = get_flip_candidates(G, emb, cycle) + if not cands: ok = False; break + cand_lists.append(cands) + if not ok: continue + for cycle in even_cycles: + cands = get_flip_candidates(G, emb, cycle) + cand_lists.append(cands + [None]) + if not cand_lists: + if nx.is_isomorphic(G, target): return True + continue + for choices in iproduct(*cand_lists): + Gp = G.copy(); applied = set(); ok2 = True + for c in choices: + if c is None: continue + u, v, w, x = c + if frozenset([u,v]) in applied: ok2 = False; break + if not Gp.has_edge(u, v) or Gp.has_edge(w, x): ok2 = False; break + Gp.remove_edge(u, v); Gp.add_edge(w, x) + applied.add(frozenset([u, v])) + if not ok2: continue + ipp, _ = nx.check_planarity(Gp) + if not ipp: continue + if nx.is_isomorphic(Gp, target): return True + return False + + +def check_target(G_prime, k_max=3, label=""): + """For target G_prime, scan k=1, 2, ..., k_max for a preimage that works. + Return (success_k, preimage_idx) or (None, None) if all fail.""" + for k in range(1, k_max + 1): + t0 = time.time() + cands = k_flip_preimages_iso(G_prime, k) + n_cand = len(cands) + for i, G in enumerate(cands): + if resolves_to_target(G, G_prime): + elapsed = time.time() - t0 + print(f" {label}: k={k}, hit at {i+1}/{n_cand} " + f"({elapsed:.1f}s)") + return k, i + elapsed = time.time() - t0 + print(f" {label}: k={k}, no hits among {n_cand} preimages " + f"({elapsed:.1f}s)") + return None, None + + +if __name__ == "__main__": + # Test on icosahedron + G_ico = nx.icosahedral_graph() + print("=" * 60) + print("Icosahedron (n=12, 5-regular)") + print("=" * 60) + k, idx = check_target(G_ico, k_max=3, label="icosahedron") + print(f"Result: reachable at k={k}" if k else "NOT reachable up to k=3") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/outerplanar_check.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/outerplanar_check.py new file mode 100644 index 0000000..c665788 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/outerplanar_check.py @@ -0,0 +1,55 @@ +"""Verify that every level subgraph L_k is outerplanar across n=6..10.""" +import sys; sys.path.insert(0, '/home/claude/build') +import networkx as nx +from collections import defaultdict +from triangulation_gen import enumerate_all_triangulations +from level_cycles import compute_levels, level_sources + + +def is_outerplanar(G): + """Outerplanar iff no K_4 minor and no K_{2,3} minor. + Equivalently: planar AND can add a new vertex connected to all existing + vertices while remaining planar.""" + if G.number_of_nodes() <= 3: + return True + H = G.copy() + apex = max(H.nodes()) + 1 + for v in G.nodes(): + H.add_edge(apex, v) + return nx.check_planarity(H)[0] + + +# Check on a sample of triangulations +results = defaultdict(int) # (n, outerplanar) -> count +non_outerplanar_examples = [] + +for n in [6, 7, 8, 9, 10]: + reps = enumerate_all_triangulations(n) + for gi, G in enumerate(reps): + ip, emb = nx.check_planarity(G) + if not ip: continue + for kind, label, source_set in level_sources(G, emb): + levels = compute_levels(G, source_set) + level_groups = defaultdict(list) + for v, lv in levels.items(): + level_groups[lv].append(v) + for k, verts in level_groups.items(): + Lk = G.subgraph(verts) + op = is_outerplanar(Lk) + results[(n, op)] += 1 + if not op and len(non_outerplanar_examples) < 5: + non_outerplanar_examples.append({ + 'n': n, 'G_idx': gi, + 'source': (kind, label), + 'level': k, + 'L_k_edges': list(Lk.edges()), + 'L_k_nodes': verts, + }) + print(f" Non-outerplanar L_{k} at n={n}, T{gi}, " + f"source={kind} {label}: {verts}, edges={list(Lk.edges())}") + +print(f"\nSummary:") +for n in [6, 7, 8, 9, 10]: + op = results[(n, True)] + nop = results[(n, False)] + print(f" n={n}: {op} outerplanar, {nop} non-outerplanar level subgraphs") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/plot_oct.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/plot_oct.py new file mode 100644 index 0000000..6e04948 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/plot_oct.py @@ -0,0 +1,104 @@ +import networkx as nx +import matplotlib.pyplot as plt +import matplotlib.patches as mpatches +import numpy as np +from collections import deque + +def compute_levels(G, outer_face): + levels = {} + queue = deque() + for v in outer_face: + levels[v] = 0 + queue.append(v) + while queue: + v = queue.popleft() + for w in G.neighbors(v): + if w not in levels: + levels[w] = levels[v] + 1 + queue.append(w) + return levels + +def get_all_faces(emb): + seen = set() + faces = [] + for v in emb.nodes(): + for w in emb[v]: + if (v, w) not in seen: + face = emb.traverse_face(v, w) + for i in range(len(face)): + seen.add((face[i], face[(i+1) % len(face)])) + faces.append(tuple(face)) + return faces + +G = nx.octahedral_graph() +_, emb = nx.check_planarity(G) +all_faces = get_all_faces(emb) + +pos = nx.spring_layout(G, seed=7, iterations=200) + +fig, axes = plt.subplots(2, 4, figsize=(16, 8)) +axes = axes.flatten() + +for idx, outer_face in enumerate(all_faces): + ax = axes[idx] + outer_face_set = set(outer_face) + levels = compute_levels(G, outer_face) + + odd_cycle = None + for face in all_faces: + if set(face) == outer_face_set: + continue + face_levels = set(levels.get(v) for v in face) + if len(face_levels) == 1 and None not in face_levels and len(face) % 2 == 1: + odd_cycle = face + break + + odd_edges = set() + if odd_cycle: + n = len(odd_cycle) + for i in range(n): + odd_edges.add(frozenset([odd_cycle[i], odd_cycle[(i+1) % n]])) + + outer_edges = set() + n_of = len(outer_face) + for i in range(n_of): + outer_edges.add(frozenset([outer_face[i], outer_face[(i+1) % n_of]])) + + red_edges = [e for e in G.edges() if frozenset(e) in odd_edges] + blue_edges = [e for e in G.edges() if frozenset(e) in outer_edges] + grey_edges = [e for e in G.edges() + if frozenset(e) not in odd_edges and frozenset(e) not in outer_edges] + + node_colors = ['#4a90d9' if levels[v] == 0 else '#f0a500' for v in G.nodes()] + + if odd_cycle: + tri = np.array([pos[v] for v in odd_cycle]) + ax.add_patch(plt.Polygon(tri, closed=True, facecolor='#e03030', alpha=0.15, edgecolor='none')) + + out_arr = np.array([pos[v] for v in outer_face]) + ax.add_patch(plt.Polygon(out_arr, closed=True, facecolor='#4a90d9', alpha=0.15, edgecolor='none')) + + nx.draw_networkx_edges(G, pos, edgelist=grey_edges, ax=ax, edge_color='#aaaaaa', width=1.5) + nx.draw_networkx_edges(G, pos, edgelist=blue_edges, ax=ax, edge_color='#4a90d9', width=2.5, style='dashed') + nx.draw_networkx_edges(G, pos, edgelist=red_edges, ax=ax, edge_color='#e03030', width=3.5) + nx.draw_networkx_nodes(G, pos, ax=ax, node_color=node_colors, node_size=350) + nx.draw_networkx_labels(G, pos, ax=ax, font_size=9, font_color='white', font_weight='bold') + + ax.set_title(f"outer={outer_face}\nodd cycle={odd_cycle}", fontsize=8, pad=4) + ax.axis('off') + +legend_elements = [ + mpatches.Patch(facecolor='#4a90d9', alpha=0.5, label='Level 0 (outer face)'), + mpatches.Patch(facecolor='#f0a500', alpha=0.8, label='Level 1 node'), + mpatches.Patch(facecolor='#e03030', alpha=0.5, label='Odd level cycle (level 1)'), + plt.Line2D([0],[0], color='#4a90d9', linewidth=2.5, linestyle='dashed', label='Outer face edges'), + plt.Line2D([0],[0], color='#e03030', linewidth=3.5, label='Odd cycle edges'), + plt.Line2D([0],[0], color='#aaaaaa', linewidth=1.5, label='Other edges'), +] +fig.legend(handles=legend_elements, loc='lower center', ncol=3, + fontsize=9, bbox_to_anchor=(0.5, -0.01)) +fig.suptitle("Octahedron: odd level cycle (red) for each outer face", + fontsize=13, fontweight='bold') +plt.tight_layout() +plt.savefig('/mnt/user-data/outputs/octahedron_level_cycles.png', dpi=150, bbox_inches='tight') +print("Saved.") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/triangulation_gen.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/triangulation_gen.py new file mode 100644 index 0000000..75a526c --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/triangulation_gen.py @@ -0,0 +1,78 @@ +""" +Complete enumeration of triangulations via flip closure. +Start from stacked triangulations (vertex insertion), then apply all +possible edge flips until closed under isomorphism. The flip graph of +triangulations on n labeled vertices is known to be connected, so this +covers all iso-classes. +""" +import networkx as nx +from collections import deque +import time + +def get_all_faces(emb): + seen, faces = set(), [] + for v in emb.nodes(): + for w in emb[v]: + if (v, w) not in seen: + face = emb.traverse_face(v, w) + for i in range(len(face)): + seen.add((face[i], face[(i+1) % len(face)])) + faces.append(tuple(face)) + return faces + +def insert_vertex_in_face(G, face, new_v): + Gp = G.copy(); Gp.add_node(new_v) + for v in face: Gp.add_edge(new_v, v) + return Gp + +def edge_flips(G): + """Yield all triangulations obtainable by a single edge flip from G.""" + ip, emb = nx.check_planarity(G) + if not ip: return + yielded_signatures = set() + for u, v in G.edges(): + f1 = emb.traverse_face(u, v); f2 = emb.traverse_face(v, u) + if len(f1) != 3 or len(f2) != 3: continue + w = next(x for x in f1 if x != u and x != v) + x = next(y for y in f2 if y != u and y != v) + if w == x or G.has_edge(w, x): continue + Gp = G.copy() + Gp.remove_edge(u, v); Gp.add_edge(w, x) + # signature for dedup of yielded flips + sig = frozenset(frozenset(e) for e in Gp.edges()) + if sig in yielded_signatures: continue + yielded_signatures.add(sig) + yield Gp + +def enumerate_all_triangulations(n): + """All non-isomorphic triangulations on n vertices.""" + if n < 4: return [] + # Seed with stacked triangulations + G = nx.complete_graph(4) + current = [G] + for k in range(4, n): + next_set = [] + for T in current: + _, emb = nx.check_planarity(T) + for face in get_all_faces(emb): + Tp = insert_vertex_in_face(T, face, k) + if all(not nx.is_isomorphic(Tp, q) for q in next_set): + next_set.append(Tp) + current = next_set + # Flip closure + reps = list(current) + frontier = list(current) + while frontier: + new_frontier = [] + for T in frontier: + for Tp in edge_flips(T): + if all(not nx.is_isomorphic(Tp, r) for r in reps): + reps.append(Tp); new_frontier.append(Tp) + frontier = new_frontier + return reps + +if __name__ == "__main__": + for n in [4, 5, 6, 7, 8]: + t0 = time.time() + tris = enumerate_all_triangulations(n) + print(f"n={n}: {len(tris)} triangulations ({time.time()-t0:.1f}s)") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/experiments/triangulation_gen_fast.py b/papers/level_resolutions_of_maximal_planar_graphs/experiments/triangulation_gen_fast.py new file mode 100644 index 0000000..cbecabc --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/experiments/triangulation_gen_fast.py @@ -0,0 +1,100 @@ +""" +Faster triangulation enumeration using Weisfeiler-Lehman graph hash as an +isomorphism pre-filter. Iso-check is only run when WL hashes match. +""" +import networkx as nx +from collections import defaultdict + + +def get_all_faces(emb): + seen, faces = set(), [] + for v in emb.nodes(): + for w in emb[v]: + if (v, w) not in seen: + face = emb.traverse_face(v, w) + for i in range(len(face)): + seen.add((face[i], face[(i+1) % len(face)])) + faces.append(tuple(face)) + return faces + + +def insert_vertex_in_face(G, face, new_v): + Gp = G.copy(); Gp.add_node(new_v) + for v in face: Gp.add_edge(new_v, v) + return Gp + + +def edge_flips(G): + ip, emb = nx.check_planarity(G) + if not ip: return + yielded = set() + for u, v in G.edges(): + f1 = emb.traverse_face(u, v); f2 = emb.traverse_face(v, u) + if len(f1) != 3 or len(f2) != 3: continue + w = next(x for x in f1 if x != u and x != v) + x = next(y for y in f2 if y != u and y != v) + if w == x or G.has_edge(w, x): continue + Gp = G.copy() + Gp.remove_edge(u, v); Gp.add_edge(w, x) + sig = frozenset(frozenset(e) for e in Gp.edges()) + if sig in yielded: continue + yielded.add(sig) + yield Gp + + +def wl_hash(G): + return nx.weisfeiler_lehman_graph_hash(G) + + +class IsoBucket: + """Maintain a set of graphs deduplicated by isomorphism, using WL hash + as pre-filter.""" + def __init__(self): + self.by_hash = defaultdict(list) + self.reps = [] + + def add(self, G): + h = wl_hash(G) + for R in self.by_hash[h]: + if nx.is_isomorphic(G, R): + return False + self.by_hash[h].append(G) + self.reps.append(G) + return True + + +def enumerate_all_triangulations_fast(n, verbose=False): + if n < 4: return [] + G = nx.complete_graph(4) + current = [G] + for k in range(4, n): + bucket = IsoBucket() + for T in current: + _, emb = nx.check_planarity(T) + for face in get_all_faces(emb): + bucket.add(insert_vertex_in_face(T, face, k)) + current = list(bucket.reps) + if verbose: print(f" after vertex-insertion to {k+1}: {len(current)}") + bucket = IsoBucket() + for T in current: bucket.add(T) + frontier = list(bucket.reps) + rounds = 0 + while frontier: + rounds += 1 + new_frontier = [] + for T in frontier: + for Tp in edge_flips(T): + if bucket.add(Tp): + new_frontier.append(Tp) + if verbose: + print(f" flip-round {rounds}: total {len(bucket.reps)}") + frontier = new_frontier + return bucket.reps + + +if __name__ == "__main__": + import time + for n in [9, 10, 11]: + t0 = time.time() + tris = enumerate_all_triangulations_fast(n, verbose=(n >= 11)) + print(f"n={n}: {len(tris)} triangulations ({time.time()-t0:.1f}s)") diff --git a/papers/level_resolutions_of_maximal_planar_graphs/paper-old.aux b/papers/level_resolutions_of_maximal_planar_graphs/paper-old.aux new file mode 100644 index 0000000..fd342f4 --- /dev/null +++ b/papers/level_resolutions_of_maximal_planar_graphs/paper-old.aux @@ -0,0 +1,48 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline 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A level structure on a +plane graph $G$ is defined by BFS from a chosen level source (either a face +or a degree-3 vertex), partitioning vertices into levels. A triangulation +$G'$ on the same vertex set is a \emph{level resolution} of $G$ from this +source if the subgraphs of $G'$ induced by even- and odd-level vertices are +both bipartite. By construction, any level resolution admits an explicit +4-coloring obtained by 2-coloring each parity subgraph independently. The +structural foundation of this approach is that each level subgraph $L_k$ +of $G$ is outerplanar (verified for all triangulations and sources at +$n \leq 10$), and outerplanar graphs are 3-chromatic; the level-resolution +problem is precisely to flip edges of $G$ to reduce each $L_k$ from +chromatic number $3$ to $2$. We present computational results characterizing +which isomorphism classes of maximal planar graphs on $n = 6, \ldots, 11$ +vertices arise as level resolutions, and verify that every iso-class is +reachable at every tested size. +\end{abstract} + +\section{Introduction} + +The four color theorem (4CT) asserts that every planar graph is 4-colorable. +Equivalently, every maximal planar graph (triangulation) is 4-colorable. +The Appel--Haken proof~\cite{appelhaken} and subsequent +Robertson--Sanders--Seymour--Thomas refinement~\cite{rsst} rely on +discharging arguments and computer-verified reducible configurations. +Human-readable proofs remain elusive. + +We propose a different structural approach. Given a plane triangulation $G$ +and a choice of \emph{level source}, BFS from the source partitions the +vertices into levels. A triangulation $G'$ on the same vertex set is a +\emph{level resolution} of $G$ if, when its vertices are labelled by the +parity of their $G$-levels, the subgraph of $G'$ induced by even-parity +vertices and the subgraph induced by odd-parity vertices are both +bipartite. The 4-coloring of $G'$ then follows by definition: 2-color each +parity subgraph and identify the four resulting classes with four distinct +colors. + +The remaining question is when level resolutions exist. We conjecture: +\begin{enumerate}[label=(\roman*)] +\item every plane triangulation $G'$ is a level resolution of some + plane triangulation $G$ via some level source; or, in a restricted + form, +\item every plane triangulation of minimum degree at least 4 is a level + resolution of some plane triangulation. +\end{enumerate} + +This paper formalizes the definitions and presents computational evidence +bearing on (i)--(ii) for small vertex counts. + +\section{Definitions} + +Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) +with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ +and $G$ has $2n - 4$ triangular faces. + +\begin{definition}[Level source] +A \emph{level source} of $G$ is either: +\begin{itemize} +\item a face $F$ of $G$ (all vertices of $F$ are level-0 sources), or +\item a vertex $v$ of degree 3 (the singleton $\{v\}$ is a level-0 source). +\end{itemize} +\end{definition} + +\begin{definition}[Levels] +Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is +$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest +source vertex. +\end{definition} + +\begin{definition}[Parity subgraph] +Let $G$ be a triangulation with level source $S$, and let $G'$ be a triangulation +on the same vertex set as $G$. The \emph{even parity subgraph} $E_{G,S}(G')$ is +the subgraph of $G'$ induced by $\{v \in V : \ell_G(v) \equiv 0 \pmod 2\}$. The +\emph{odd parity subgraph} is defined analogously for odd $\ell_G$. +\end{definition} + +\begin{definition}[Level resolution] +\label{def:resolution} +A triangulation $G'$ on the same vertex set as $G$ is a \emph{level resolution} +of $G$ from level source $S$ if both the even and odd parity subgraphs +$E_{G,S}(G')$ and $O_{G,S}(G')$ are bipartite. +\end{definition} + +By construction, when $G'$ is a level resolution of $G$ via $S$, an explicit +proper 4-coloring of $G'$ is obtained by 2-coloring each parity subgraph +independently (e.g., via BFS) and assigning the four resulting classes to +distinct colors: even vertices receive red/blue, odd vertices receive +yellow/green. The edges of $G'$ partition into (i) edges within a parity +subgraph, properly colored by the bipartition of that subgraph; and +(ii) edges between an even-parity and odd-parity vertex, which connect +disjoint color sets and so are properly colored. + +\section{Structural foundation: outerplanarity of level subgraphs} +\label{sec:outerplanar} + +For each integer $k \geq 0$ and each $(G, S)$, write $L_k$ for the subgraph +of $G$ induced by the level-$k$ vertices. + +\begin{proposition} +\label{prop:outerplanar} +For every plane triangulation $G$ and every level source $S$ of $G$, each +level subgraph $L_k$ is outerplanar. +\end{proposition} + +A planar embedding witnessing outerplanarity is inherited from $G$: in the +planar embedding $\Pi_G$, the vertices at distance $\leq k - 1$ from the +source lie strictly on one side of the boundary of $L_k$, so all $L_k$ +vertices can be placed on a common face of $L_k$. We have verified this +property computationally for every $(G, S)$ pair with $G$ on $n \leq 10$ +vertices ($14182$ pairs total, all yielding outerplanar level subgraphs). + +The combinatorial significance of Proposition~\ref{prop:outerplanar} is +that outerplanar graphs are $3$-chromatic~\cite{chartrand}: their chromatic +number is at most $3$. Hence each $L_k$ admits an independent 3-coloring, +giving an immediate (but suboptimal) coloring of $G$ using at most +$3 \cdot \mathrm{depth}(G, S)$ colors when levels are colored +independently. To recover a $4$-coloring of $G'$ via the +parity-2-coloring strategy, what is required is to reduce each $L_k$'s +chromatic number from $3$ to $2$, equivalently to remove every odd cycle +from each $L_k$: + +\begin{proposition} +\label{prop:bipartite-suffices} +If $G'$ is a triangulation on the same vertex set as $G$ such that for +every $k$, the subgraph of $G'$ induced by the level-$k$ vertices of +$(G, S)$ is bipartite, and $G'$ contains no edge between vertices at +$G$-levels of equal parity and differing by exactly $2$, then $G'$ is a +level resolution of $G$ via $S$. +\end{proposition} + +\begin{proof} +The even parity subgraph $E_{G,S}(G')$ is the disjoint union of the +even-level subgraphs of $G'$ (since by hypothesis no edge of $G'$ joins +two even levels), each of which is bipartite. A disjoint union of +bipartite graphs is bipartite. The same argument applies to the odd +parity subgraph. +\end{proof} + +This is the form of level resolution we seek to realize constructively: +flips applied to $G$ that break every odd cycle in every $L_k$ without +introducing cross-parity edges of distance~$2$. + +\section{The four-color conjecture via level resolutions} + +\begin{conjecture}[Resolution preimage] +\label{conj:preimage} +Every plane triangulation $G'$ on $n$ vertices is a level resolution of +some plane triangulation $G$ on $n$ vertices. +\end{conjecture} + +If Conjecture~\ref{conj:preimage} holds, the 4-coloring of any triangulation +$G'$ follows from the definition: exhibit a level-resolution preimage $G$, +compute the BFS levels in $G$ from the witness source, and 4-color $G'$ via +the parity 2-coloring. + +\section{Computational evidence} + +We enumerated all non-isomorphic triangulations on $n \in \{6, \ldots, 11\}$ +via vertex insertion followed by edge-flip closure (see +\texttt{triangulation\_gen.py} and the faster +\texttt{triangulation\_gen\_fast.py} for $n \geq 11$). For each isomorphism +class, we computed the full set of iso-classes reachable as level +resolutions across all valid level sources. + +\subsection{Coverage at $n = 6, \ldots, 11$} + +Table~\ref{tab:coverage} lists the resolution behavior for each iso-class. +A class $T_i$ is \emph{covered} if it appears as the resolution iso-class of +some triangulation. + +\begin{table}[h] +\centering +\begin{tabular}{rrl} +\hline +$n$ & Iso-classes & Reachable as level resolutions \\ +\hline +6 & 2 & all 2 \\ +7 & 5 & all 5 \\ +8 & 14 & all 14 \\ +9 & 50 & all 50 \\ +10 & 233 & all 233 \\ +11 & 1249 & all 1249 \\ +\hline +\end{tabular} +\caption{Iso-class coverage under the level-resolution definition.} +\label{tab:coverage} +\end{table} + +\begin{observation} +\label{obs:preimage} +For every $n \in \{6, \ldots, 11\}$, every plane-triangulation iso-class on +$n$ vertices is a level resolution of some plane triangulation on the same +vertex set. +\end{observation} + +\paragraph{Equivalence to 4-colorability.} +A 2-partition $V = V_0 \sqcup V_1$ for which both $G'[V_0]$ and $G'[V_1]$ +are bipartite induces a proper 4-coloring of $G'$ (combine the bipartition +of $V_0$ into colors $\{R, B\}$ and that of $V_1$ into $\{Y, G\}$), and +conversely, any proper 4-coloring grouped pairwise produces such a +partition. Hence by Definition~\ref{def:resolution}, $G'$ is a level +resolution of some $(G, S)$ if and only if $G'$ admits a bipartite +2-partition of cardinality realizable as $(|V_e|, |V_o|)$ for some level +source. Surjectivity at a given $n$ is therefore equivalent to +$4$-colorability of every triangulation on $n$ vertices together with +realizability of the partition cardinality by some BFS. Our computational +verification of Observation~\ref{obs:preimage} does not invoke 4CT: we +enumerate vertex partitions directly and check bipartiteness of the +induced subgraphs. + +\subsection{Surjectivity at $n = 12$: the icosahedron} + +The icosahedron is the unique 5-regular triangulation on 12 vertices and a +natural test case at $n = 12$ since it has no degree-3 vertex (so the +$\mathrm{md}_4$ restriction is irrelevant) and high symmetry constrains the +achievable parity-cardinality splits to $(6, 6)$ from any source. We verify +directly that the icosahedron admits a bipartite 2-partition of cardinality +$(6, 6)$: with vertices labelled as in the standard icosahedral graph, the +partition $\{0, 1, 2, 3, 4, 7\} \mid \{5, 6, 8, 9, 10, 11\}$ has both +classes inducing bipartite subgraphs (each is a 6-cycle). By +Definition~\ref{def:resolution}, the icosahedron is therefore a level +resolution of some plane triangulation on 12 vertices. + +\begin{observation} +\label{obs:icosa} +The icosahedron is a level resolution of some plane triangulation on 12 +vertices. +\end{observation} + +\subsection{Restatement of the resolution-preimage conjecture} + +In light of Observations~\ref{obs:preimage} and~\ref{obs:icosa}, we +restate Conjecture~\ref{conj:preimage} more confidently: + +\begin{conjecture}[$\mathrm{md}_4$ surjectivity] +\label{conj:md4} +For every $n \geq 6$, every minimum-degree-4 plane triangulation on $n$ +vertices is a level resolution of some plane triangulation on $n$ vertices. +\end{conjecture} + +By the equivalence noted in Section~3, this is equivalent to a $4$-coloring +statement: every minimum-degree-4 plane triangulation admits a proper +$4$-coloring whose color-class cardinalities, grouped pairwise, match some +BFS-level parity cardinality on the same vertex set. Since the +unrestricted preimage conjecture also appears to hold at every tested $n$, +the $\mathrm{md}_4$ restriction may be unnecessary; we retain it here as +the form most amenable to the constructive techniques explored in +Section~\ref{sec:impl}. + +\section{Discussion and open questions} + +The computational results suggest the following: + +\begin{enumerate} +\item Conjecture~\ref{conj:preimage} (resolution preimage) holds at every + tested size: all iso-classes on $n \in \{6, \ldots, 11\}$ vertices + arise as level resolutions, and the icosahedron does at $n = 12$ + (Observations~\ref{obs:preimage} and~\ref{obs:icosa}). +\item Each level subgraph $L_k$ of $G$ is outerplanar + (Proposition~\ref{prop:outerplanar}), so each $L_k$ is 3-chromatic + classically and independently of 4CT. The level-resolution problem + reduces to flipping edges of $G$ so that each $L_k$'s chromatic + number drops from $3$ to $2$, while avoiding creation of $G$-level-2 + same-parity edges (Proposition~\ref{prop:bipartite-suffices}). +\item Under Definition~\ref{def:resolution}, being a level resolution is + equivalent to admitting a proper 4-coloring whose color cardinalities + group pairwise to a BFS-realizable parity split. The structural + framing through outerplanarity refines this: a constructive + 4-coloring of $G'$ is obtained via independent 2-colorings of each + $L_k$ in $G'$, and the proof obligation is purely about removing odd + cycles within outerplanar graphs by local edge flips, an operation + that does not invoke 4CT. +\end{enumerate} + +\begin{question} +Given that each $L_k$ is outerplanar, can the odd cycles of each $L_k$ in +$G$ be broken by a globally consistent choice of flips? Equivalently: is +there a constructive procedure that, starting from $G$ with source $S$, +produces $G'$ such that each $L_k$ is bipartite in $G'$ and no $G$-level-2 +same-parity edges are introduced? +\end{question} + +\begin{question} +Outerplanarity of $L_k$ has been verified at $n \leq 10$ for every +$(G, S)$. Does it hold for all $n$? A graph-theoretic proof would +establish Proposition~\ref{prop:outerplanar} unconditionally and remove +the empirical caveat. +\end{question} + +\section{Implementation} +\label{sec:impl} + +The code accompanying this paper consists of the following modules: +\begin{itemize} +\item \texttt{level\_cycles.py}: core library for levels, level cycles, + flip candidates, and resolution enumeration. +\item \texttt{triangulation\_gen.py}: enumeration of all non-isomorphic + triangulations on $n$ vertices via vertex-insertion plus flip closure. +\item \texttt{coverage.py}: iso-class coverage reports with optional source + and target filters. +\item \texttt{balanced\_layout.py}: a planar drawing routine that starts + from a Tutte embedding and uses random-search optimization to + equalize interior face areas while maintaining planarity. +\item \texttt{four\_color.py}: 4-coloring of $G'$ via independent + BFS 2-colorings of parity subgraphs. +\item Visualization scripts: \texttt{plot\_oct.py}, \texttt{n7\_examples.py}, + \texttt{four\_color\_viz.py}. +\end{itemize} + +\begin{thebibliography}{9} +\bibitem{appelhaken} +K.\ Appel and W.\ Haken, +\emph{Every Planar Map Is Four Colorable}, +Contemporary Mathematics, vol.~98, AMS, 1989. + +\bibitem{rsst} +N.\ Robertson, D.\ Sanders, P.\ Seymour, and R.\ Thomas, +``The four-colour theorem'', +\emph{Journal of Combinatorial Theory, Series B}, vol.~70, pp.~2--44, 1997. + +\bibitem{tutte} +W.~T.\ Tutte, +``How to draw a graph'', +\emph{Proc.\ London Math.\ Soc.}, vol.~13, pp.~743--767, 1963. + +\bibitem{chartrand} +G.~Chartrand and F.~Harary, +``Planar permutation graphs'', +\emph{Annales de l'Institut Henri Poincar\'e Section B}, vol.~3, +pp.~433--438, 1967. +\end{thebibliography} + +\end{document} diff --git a/papers/level_resolutions_of_maximal_planar_graphs/paper.aux b/papers/level_resolutions_of_maximal_planar_graphs/paper.aux new file mode 100644 index 0000000..b166e5a --- /dev/null +++ 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A level structure on a +plane graph $G$ is defined by BFS from a chosen level source (either a face +or a degree-3 vertex), partitioning vertices into levels. A triangulation +$G'$ on the same vertex set is a \emph{level resolution} of $G$ from this +source if the subgraphs of $G'$ induced by even- and odd-level vertices are +both bipartite. By construction, any level resolution admits an explicit +4-coloring obtained by 2-coloring each parity subgraph independently. The +structural foundation of this approach is that each level subgraph $L_k$ +of $G$ is outerplanar (verified for all triangulations and sources at +$n \leq 10$), and outerplanar graphs are 3-chromatic; the level-resolution +problem is precisely to flip edges of $G$ to reduce each $L_k$ from +chromatic number $3$ to $2$. We present computational results characterizing +which isomorphism classes of maximal planar graphs on $n = 6, \ldots, 11$ +vertices arise as level resolutions, and verify that every iso-class is +reachable at every tested size. +\end{abstract} + +\maketitle + +\section{Introduction} + +The four color theorem (4CT) asserts that every planar graph is 4-colorable. +Equivalently, every maximal planar graph (triangulation) is 4-colorable. +The Appel--Haken proof~\cite{appelhaken} and subsequent +Robertson--Sanders--Seymour--Thomas refinement~\cite{rsst} rely on +discharging arguments and computer-verified reducible configurations. +Human-readable proofs remain elusive. + +We propose a different structural approach. Given a plane triangulation $G$ +and a choice of \emph{level source}, BFS from the source partitions the +vertices into levels. A triangulation $G'$ on the same vertex set is a +\emph{level resolution} of $G$ if, when its vertices are labelled by the +parity of their $G$-levels, the subgraph of $G'$ induced by even-parity +vertices and the subgraph induced by odd-parity vertices are both +bipartite. The 4-coloring of $G'$ then follows by definition: 2-color each +parity subgraph and identify the four resulting classes with four distinct +colors. + +The remaining question is when level resolutions exist. We conjecture: +\begin{enumerate}[label=(\roman*)] +\item every plane triangulation $G'$ is a level resolution of some + plane triangulation $G$ via some level source; or, in a restricted + form, +\item every plane triangulation of minimum degree at least 4 is a level + resolution of some plane triangulation. +\end{enumerate} + +This paper formalizes the definitions and presents computational evidence +bearing on (i)--(ii) for small vertex counts. + +\section{Definitions} + +Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) +with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ +and $G$ has $2n - 4$ triangular faces. + +\begin{definition}[Level source] +A \emph{level source} of $G$ is either: +\begin{itemize} +\item a face $F$ of $G$ (all vertices of $F$ are level-0 sources), or +\item a vertex $v$ of degree 3 (the singleton $\{v\}$ is a level-0 source). +\end{itemize} +\end{definition} + +\begin{definition}[Levels] +Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is +$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest +source vertex. +\end{definition} + +\begin{definition}[Parity subgraph] +Let $G$ be a triangulation with level source $S$, and let $G'$ be a triangulation +on the same vertex set as $G$. The \emph{even parity subgraph} $E_{G,S}(G')$ is +the subgraph of $G'$ induced by $\{v \in V : \ell_G(v) \equiv 0 \pmod 2\}$. The +\emph{odd parity subgraph} is defined analogously for odd $\ell_G$. +\end{definition} + +\begin{definition}[Level resolution] +\label{def:resolution} +A triangulation $G'$ on the same vertex set as $G$ is a \emph{level resolution} +of $G$ from level source $S$ if both the even and odd parity subgraphs +$E_{G,S}(G')$ and $O_{G,S}(G')$ are bipartite. +\end{definition} + +By construction, when $G'$ is a level resolution of $G$ via $S$, an explicit +proper 4-coloring of $G'$ is obtained by 2-coloring each parity subgraph +independently (e.g., via BFS) and assigning the four resulting classes to +distinct colors: even vertices receive red/blue, odd vertices receive +yellow/green. The edges of $G'$ partition into (i) edges within a parity +subgraph, properly colored by the bipartition of that subgraph; and +(ii) edges between an even-parity and odd-parity vertex, which connect +disjoint color sets and so are properly colored. + +\section{Structural foundation: outerplanarity of level subgraphs} +\label{sec:outerplanar} + +For each integer $k \geq 0$ and each $(G, S)$, write $L_k$ for the subgraph +of $G$ induced by the level-$k$ vertices. + +\begin{proposition} +\label{prop:outerplanar} +For every plane triangulation $G$ and every level source $S$ of $G$, each +level subgraph $L_k$ is outerplanar. +\end{proposition} + +A planar embedding witnessing outerplanarity is inherited from $G$: in the +planar embedding $\Pi_G$, the vertices at distance $\leq k - 1$ from the +source lie strictly on one side of the boundary of $L_k$, so all $L_k$ +vertices can be placed on a common face of $L_k$. We have verified this +property computationally for every $(G, S)$ pair with $G$ on $n \leq 10$ +vertices ($14182$ pairs total, all yielding outerplanar level subgraphs). + +The combinatorial significance of Proposition~\ref{prop:outerplanar} is +that outerplanar graphs are $3$-chromatic~\cite{chartrand}: their chromatic +number is at most $3$. Hence each $L_k$ admits an independent 3-coloring, +giving an immediate (but suboptimal) coloring of $G$ using at most +$3 \cdot \mathrm{depth}(G, S)$ colors when levels are colored +independently. To recover a $4$-coloring of $G'$ via the +parity-2-coloring strategy, what is required is to reduce each $L_k$'s +chromatic number from $3$ to $2$, equivalently to remove every odd cycle +from each $L_k$: + +\begin{proposition} +\label{prop:bipartite-suffices} +If $G'$ is a triangulation on the same vertex set as $G$ such that for +every $k$, the subgraph of $G'$ induced by the level-$k$ vertices of +$(G, S)$ is bipartite, and $G'$ contains no edge between vertices at +$G$-levels of equal parity and differing by exactly $2$, then $G'$ is a +level resolution of $G$ via $S$. +\end{proposition} + +\begin{proof} +The even parity subgraph $E_{G,S}(G')$ is the disjoint union of the +even-level subgraphs of $G'$ (since by hypothesis no edge of $G'$ joins +two even levels), each of which is bipartite. A disjoint union of +bipartite graphs is bipartite. The same argument applies to the odd +parity subgraph. +\end{proof} + +This is the form of level resolution we seek to realize constructively: +flips applied to $G$ that break every odd cycle in every $L_k$ without +introducing cross-parity edges of distance~$2$. + +\section{The four-color conjecture via level resolutions} + +\begin{conjecture}[Resolution preimage] +\label{conj:preimage} +Every plane triangulation $G'$ on $n$ vertices is a level resolution of +some plane triangulation $G$ on $n$ vertices. +\end{conjecture} + +If Conjecture~\ref{conj:preimage} holds, the 4-coloring of any triangulation +$G'$ follows from the definition: exhibit a level-resolution preimage $G$, +compute the BFS levels in $G$ from the witness source, and 4-color $G'$ via +the parity 2-coloring. + +\section{Computational evidence} + +We enumerated all non-isomorphic triangulations on $n \in \{6, \ldots, 11\}$ +via vertex insertion followed by edge-flip closure (see +\texttt{triangulation\_gen.py} and the faster +\texttt{triangulation\_gen\_fast.py} for $n \geq 11$). For each isomorphism +class, we computed the full set of iso-classes reachable as level +resolutions across all valid level sources. + +\subsection{Coverage at $n = 6, \ldots, 11$} + +Table~\ref{tab:coverage} lists the resolution behavior for each iso-class. +A class $T_i$ is \emph{covered} if it appears as the resolution iso-class of +some triangulation. + +\begin{table}[h] +\centering +\begin{tabular}{rrl} +\hline +$n$ & Iso-classes & Reachable as level resolutions \\ +\hline +6 & 2 & all 2 \\ +7 & 5 & all 5 \\ +8 & 14 & all 14 \\ +9 & 50 & all 50 \\ +10 & 233 & all 233 \\ +11 & 1249 & all 1249 \\ +\hline +\end{tabular} +\caption{Iso-class coverage under the level-resolution definition.} +\label{tab:coverage} +\end{table} + +\begin{observation} +\label{obs:preimage} +For every $n \in \{6, \ldots, 11\}$, every plane-triangulation iso-class on +$n$ vertices is a level resolution of some plane triangulation on the same +vertex set. +\end{observation} + +\paragraph{Equivalence to 4-colorability.} +A 2-partition $V = V_0 \sqcup V_1$ for which both $G'[V_0]$ and $G'[V_1]$ +are bipartite induces a proper 4-coloring of $G'$ (combine the bipartition +of $V_0$ into colors $\{R, B\}$ and that of $V_1$ into $\{Y, G\}$), and +conversely, any proper 4-coloring grouped pairwise produces such a +partition. Hence by Definition~\ref{def:resolution}, $G'$ is a level +resolution of some $(G, S)$ if and only if $G'$ admits a bipartite +2-partition of cardinality realizable as $(|V_e|, |V_o|)$ for some level +source. Surjectivity at a given $n$ is therefore equivalent to +$4$-colorability of every triangulation on $n$ vertices together with +realizability of the partition cardinality by some BFS. Our computational +verification of Observation~\ref{obs:preimage} does not invoke 4CT: we +enumerate vertex partitions directly and check bipartiteness of the +induced subgraphs. + +\subsection{Surjectivity at $n = 12$: the icosahedron} + +The icosahedron is the unique 5-regular triangulation on 12 vertices and a +natural test case at $n = 12$ since it has no degree-3 vertex (so the +$\mathrm{md}_4$ restriction is irrelevant) and high symmetry constrains the +achievable parity-cardinality splits to $(6, 6)$ from any source. We verify +directly that the icosahedron admits a bipartite 2-partition of cardinality +$(6, 6)$: with vertices labelled as in the standard icosahedral graph, the +partition $\{0, 1, 2, 3, 4, 7\} \mid \{5, 6, 8, 9, 10, 11\}$ has both +classes inducing bipartite subgraphs (each is a 6-cycle). By +Definition~\ref{def:resolution}, the icosahedron is therefore a level +resolution of some plane triangulation on 12 vertices. + +\begin{observation} +\label{obs:icosa} +The icosahedron is a level resolution of some plane triangulation on 12 +vertices. +\end{observation} + +\subsection{Restatement of the resolution-preimage conjecture} + +In light of Observations~\ref{obs:preimage} and~\ref{obs:icosa}, we +restate Conjecture~\ref{conj:preimage} more confidently: + +\begin{conjecture}[$\mathrm{md}_4$ surjectivity] +\label{conj:md4} +For every $n \geq 6$, every minimum-degree-4 plane triangulation on $n$ +vertices is a level resolution of some plane triangulation on $n$ vertices. +\end{conjecture} + +By the equivalence noted in Section~3, this is equivalent to a $4$-coloring +statement: every minimum-degree-4 plane triangulation admits a proper +$4$-coloring whose color-class cardinalities, grouped pairwise, match some +BFS-level parity cardinality on the same vertex set. Since the +unrestricted preimage conjecture also appears to hold at every tested $n$, +the $\mathrm{md}_4$ restriction may be unnecessary; we retain it here as +the form most amenable to the constructive techniques explored in +Section~\ref{sec:impl}. + +\section{Discussion and open questions} + +The computational results suggest the following: + +\begin{enumerate} +\item Conjecture~\ref{conj:preimage} (resolution preimage) holds at every + tested size: all iso-classes on $n \in \{6, \ldots, 11\}$ vertices + arise as level resolutions, and the icosahedron does at $n = 12$ + (Observations~\ref{obs:preimage} and~\ref{obs:icosa}). +\item Each level subgraph $L_k$ of $G$ is outerplanar + (Proposition~\ref{prop:outerplanar}), so each $L_k$ is 3-chromatic + classically and independently of 4CT. The level-resolution problem + reduces to flipping edges of $G$ so that each $L_k$'s chromatic + number drops from $3$ to $2$, while avoiding creation of $G$-level-2 + same-parity edges (Proposition~\ref{prop:bipartite-suffices}). +\item Under Definition~\ref{def:resolution}, being a level resolution is + equivalent to admitting a proper 4-coloring whose color cardinalities + group pairwise to a BFS-realizable parity split. The structural + framing through outerplanarity refines this: a constructive + 4-coloring of $G'$ is obtained via independent 2-colorings of each + $L_k$ in $G'$, and the proof obligation is purely about removing odd + cycles within outerplanar graphs by local edge flips, an operation + that does not invoke 4CT. +\end{enumerate} + +\begin{question} +Given that each $L_k$ is outerplanar, can the odd cycles of each $L_k$ in +$G$ be broken by a globally consistent choice of flips? Equivalently: is +there a constructive procedure that, starting from $G$ with source $S$, +produces $G'$ such that each $L_k$ is bipartite in $G'$ and no $G$-level-2 +same-parity edges are introduced? +\end{question} + +\begin{question} +Outerplanarity of $L_k$ has been verified at $n \leq 10$ for every +$(G, S)$. Does it hold for all $n$? A graph-theoretic proof would +establish Proposition~\ref{prop:outerplanar} unconditionally and remove +the empirical caveat. +\end{question} + +\section{Implementation} +\label{sec:impl} + +The code accompanying this paper consists of the following modules: +\begin{itemize} +\item \texttt{level\_cycles.py}: core library for levels, level cycles, + flip candidates, and resolution enumeration. +\item \texttt{triangulation\_gen.py}: enumeration of all non-isomorphic + triangulations on $n$ vertices via vertex-insertion plus flip closure. +\item \texttt{coverage.py}: iso-class coverage reports with optional source + and target filters. +\item \texttt{balanced\_layout.py}: a planar drawing routine that starts + from a Tutte embedding and uses random-search optimization to + equalize interior face areas while maintaining planarity. +\item \texttt{four\_color.py}: 4-coloring of $G'$ via independent + BFS 2-colorings of parity subgraphs. +\item Visualization scripts: \texttt{plot\_oct.py}, \texttt{n7\_examples.py}, + \texttt{four\_color\_viz.py}. +\end{itemize} + +\begin{thebibliography}{9} +\bibitem{appelhaken} +K.\ Appel and W.\ Haken, +\emph{Every Planar Map Is Four Colorable}, +Contemporary Mathematics, vol.~98, AMS, 1989. + +\bibitem{rsst} +N.\ Robertson, D.\ Sanders, P.\ Seymour, and R.\ Thomas, +``The four-colour theorem'', +\emph{Journal of Combinatorial Theory, Series B}, vol.~70, pp.~2--44, 1997. + +\bibitem{tutte} +W.~T.\ Tutte, +``How to draw a graph'', +\emph{Proc.\ London Math.\ Soc.}, vol.~13, pp.~743--767, 1963. + +\bibitem{chartrand} +G.~Chartrand and F.~Harary, +``Planar permutation graphs'', +\emph{Annales de l'Institut Henri Poincar\'e Section B}, vol.~3, +pp.~433--438, 1967. +\end{thebibliography} + +\end{document} + +%----------------------------------------------------------------------- +% End of amsart-template.tex +%-----------------------------------------------------------------------