From 10a53e9de126bffc169d1cd7486dd889811e6022 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 03:42:08 -0400 Subject: [PATCH] =?UTF-8?q?face=5Fmonochromatic=5Fpairs:=20strengthen=20Co?= =?UTF-8?q?nj=205.5=20to=20face-length=20=E2=89=A5=205,=20find=2028-vertex?= =?UTF-8?q?=20counterexample?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - Paper: Conjecture 5.5 restated with the hypothesis that every face of H has length ≥ 5 (= the cubic plane analogue of "no triangles or quadrilaterals as faces"). This kills K_4 and the n=8 trivial counterexamples (girth 3) and the ad-hoc n=40 counterexample (which has 2 triangles and 4 quadrilaterals). A new remark catalogues these excluded counterexamples and the smallest cubic plane graphs satisfying the hypothesis (dodecahedron at |V| = 20). - search_smaller_counterexample.py: --min-face=N option to filter cubic planar graphs by minimum face length. - search_min_face5_counterexample.py: enumerates triangulations T with min degree ≥ 5 via graphs.triangulations(n, minimum_degree=5), takes planar dual (= cubic plane with all faces ≥ 5), and runs the Heawood-constancy check. - Result: smallest counterexample at triangulation order n_T = 16, whose dual is a 28-vertex cubic plane graph (graph6 [kG[A?_A?_?_?K?D?@_CO?o?@_??A??@C??O??AG?C????`???a???W???A_???F). Faces: 12 pentagons + 4 hexagons (a C28 fullerene). Both K_{red, blue} and K_{red, green} are 12-cycles sharing the colour-red edge (0, 1) and both have h_φ ≡ -1. 8 of 28 vertices lie outside V(K_0) ∪ V(K_1). - verify_28_vertex_counterexample.py: reproduces the counterexample, verifies all properties, and renders figures/min-face-5-counterexample.png. Note on the boundary: face-length ≥ 6 is impossible for cubic plane graphs by Euler (6F = 6(V/2 + 2) > 3V = sum face lengths for V > 4). So face-length ≥ 5 is the strongest face-length restriction admitting any cubic plane graphs at all. Co-Authored-By: Claude Opus 4.7 --- .../search_min_face5_counterexample.py | 137 ++++++++++++++++++ .../search_smaller_counterexample.py | 41 +++++- .../verify_28_vertex_counterexample.py | 120 +++++++++++++++ .../figures/min-face-5-counterexample.png | Bin 0 -> 192060 bytes papers/face_monochromatic_pairs/paper.pdf | Bin 1202293 -> 1192277 bytes papers/face_monochromatic_pairs/paper.tex | 97 ++++++------- 6 files changed, 342 insertions(+), 53 deletions(-) create mode 100644 papers/face_monochromatic_pairs/experiments/search_min_face5_counterexample.py create mode 100644 papers/face_monochromatic_pairs/experiments/verify_28_vertex_counterexample.py create mode 100644 papers/face_monochromatic_pairs/figures/min-face-5-counterexample.png diff --git a/papers/face_monochromatic_pairs/experiments/search_min_face5_counterexample.py b/papers/face_monochromatic_pairs/experiments/search_min_face5_counterexample.py new file mode 100644 index 0000000..1d30765 --- /dev/null +++ b/papers/face_monochromatic_pairs/experiments/search_min_face5_counterexample.py @@ -0,0 +1,137 @@ +"""Search for counterexamples to the strengthened Conjecture 5.5: + + Let H be a cubic plane graph in which every face has length ≥ 5, + with a proper 3-edge-colouring φ. If K_0 = {a,b}-Kempe cycle and + K_1 = {a,c}-Kempe cycle share a colour-a edge, then h_φ cannot be + constant on both V(K_0) and V(K_1). + +The graphs H satisfying the face-length-≥-5 hypothesis are exactly +the duals of triangulations of the sphere with minimum degree ≥ 5. +We enumerate these via plantri (through Sage's +`graphs.triangulations(n, minimum_degree=5)`), take the planar dual, +and run the same Heawood-constancy check as +`search_smaller_counterexample.py`. + +For each triangulation order n, the dual has 2n - 4 cubic vertices. +The smallest case n = 12 is the icosahedron, whose dual is the +dodecahedron (20 vertices, all faces pentagonal). + +Run with: + sage experiments/search_min_face5_counterexample.py # default max_n=14 + sage experiments/search_min_face5_counterexample.py 16 + sage experiments/search_min_face5_counterexample.py 16 --all +""" +import sys +import time + +from sage.all import Graph +from sage.graphs.graph_generators import graphs + +# Reuse the verification machinery from the cubic-search script. +import os +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +from search_smaller_counterexample import ( + check_graph, + min_face_length, +) + + +def planar_dual(G): + """Build the planar dual of a planar graph G with its embedding + already set via G.is_planar(set_embedding=True).""" + faces = G.faces() + D = Graph(multiedges=False, loops=False) + n_faces = len(faces) + D.add_vertices(range(n_faces)) + # Map each (undirected) edge of G to the indices of the two faces + # that contain it. + edge_to_faces = {} + for i, face in enumerate(faces): + for e in face: + key = frozenset(e[:2]) + edge_to_faces.setdefault(key, []).append(i) + for key, fs in edge_to_faces.items(): + if len(fs) == 2 and fs[0] != fs[1]: + D.add_edge(fs[0], fs[1]) + return D + + +def main(): + args = [a for a in sys.argv[1:] if not a.startswith('--')] + max_n = int(args[0]) if args else 14 + flag_all = '--all' in sys.argv[1:] + + print(f"Searching for counterexamples to the face-length-≥-5 form " + f"of Conjecture 5.5.\n" + f"Iterating over triangulations with min degree ≥ 5 for " + f"n_T in [12, {max_n}].\n" + f"Each dual is cubic with all faces of length ≥ 5; the dual " + f"has 2n_T - 4 vertices.\n" + f"{'Continuing past first hit.' if flag_all else 'Stopping at first hit.'}\n") + + first_found = None + for n_T in range(12, max_n + 1): + start = time.time() + count = 0 + found = None + try: + gen = graphs.triangulations(n_T, minimum_degree=5) + except Exception as ex: + print(f"n_T={n_T:>3}: cannot enumerate ({ex})") + continue + for T in gen: + T.is_planar(set_embedding=True) + H = planar_dual(T) + n_H = H.order() + # Sanity: H should be cubic and planar; faces length ≥ 5. + if max(H.degree()) != 3 or min(H.degree()) != 3: + continue + if not H.is_planar(set_embedding=True): + continue + if min_face_length(H) < 5: + continue + count += 1 + res = check_graph(H) + if res is not None: + found = (T.copy(), H.copy(), res, n_H) + if not flag_all: + break + elapsed = time.time() - start + n_H_min = 2 * n_T - 4 + if found is None: + print(f"n_T={n_T:>3}: checked {count} triangulation duals " + f"(each with {n_H_min} cubic vertices), no counterexample " + f"[{elapsed:.1f}s]") + else: + T, H, (col, K0, K1, h0, h1, a, b, c, e), n_H = found + colour_name = {0: 'red', 1: 'blue', 2: 'green'} + print(f"n_T={n_T:>3}: COUNTEREXAMPLE in dual #{count} " + f"(|V(H)| = {n_H}) [{elapsed:.1f}s]") + print(f" triangulation canonical graph6 = " + f"{T.canonical_label().graph6_string()}") + print(f" dual (cubic) canonical graph6 = " + f"{H.canonical_label().graph6_string()}") + print(f" dual edges = {sorted(H.edges(labels=False))}") + print(f" colouring = {col}") + print(f" shared colour = {colour_name[a]} ({a}), edge {e}") + print(f" K_{{a,b}} = K_{{{colour_name[a]},{colour_name[b]}}} " + f"= {K0} (h={h0:+d}, |V|={len(K0)})") + print(f" K_{{a,c}} = K_{{{colour_name[a]},{colour_name[c]}}} " + f"= {K1} (h={h1:+d}, |V|={len(K1)})") + if first_found is None: + first_found = n_T + if not flag_all: + break + sys.stdout.flush() + + if first_found is not None: + print(f"\nSmallest counterexample at triangulation order n_T " + f"= {first_found} (dual has {2 * first_found - 4} vertices).") + else: + print(f"\nNo counterexample found for triangulation orders ≤ " + f"{max_n} (dual orders ≤ {2 * max_n - 4}).") + + +if __name__ == '__main__': + main() diff --git a/papers/face_monochromatic_pairs/experiments/search_smaller_counterexample.py b/papers/face_monochromatic_pairs/experiments/search_smaller_counterexample.py index f1ac02b..c583b35 100644 --- a/papers/face_monochromatic_pairs/experiments/search_smaller_counterexample.py +++ b/papers/face_monochromatic_pairs/experiments/search_smaller_counterexample.py @@ -146,19 +146,45 @@ def check_graph(G): return None +def min_face_length(G): + """Min face length of G in its current planar embedding.""" + return min(len(f) for f in G.faces()) + + def main(): args = [a for a in sys.argv[1:] if not a.startswith('--')] max_n = int(args[0]) if args else 18 flag_all = '--all' in sys.argv[1:] + # Parse --min-face N + min_face = None + for i, a in enumerate(sys.argv[1:]): + if a == '--min-face' and i + 1 < len(sys.argv) - 1: + min_face = int(sys.argv[i + 2]) + break + if a.startswith('--min-face='): + min_face = int(a.split('=', 1)[1]) + break print(f"Searching for cubic plane graph counterexamples to " f"Conjecture 5.5, n in [4, {max_n}] " - f"({'continuing past first hit' if flag_all else 'stopping at first hit'})\n") + f"({'continuing past first hit' if flag_all else 'stopping at first hit'})" + + (f", min face length >= {min_face}" if min_face else "") + + "\n") + + # If filtering by min face length, skip small n where impossible. + # For all-faces-length->=L cubic plane: V - E + F = 2, E = 3V/2, + # sum face lengths = 3V, F = V/2 + 2, so min sum = L*(V/2 + 2) + # <= 3V gives V >= 2L*(L-3)/(3-L/2)... let's just set V >= 20 for L=5. + if min_face is not None and min_face >= 5: + n_start = max(4, 20) + else: + n_start = 4 first_found = None - for n in range(4, max_n + 1, 2): # cubic requires n even + for n in range(n_start, max_n + 1, 2): # cubic requires n even start = time.time() count = 0 + skipped = 0 found = None try: gen = graphs.planar_graphs( @@ -172,6 +198,11 @@ def main(): for G in gen: if max(G.degree()) != 3: continue # not cubic + if min_face is not None: + G.is_planar(set_embedding=True) + if min_face_length(G) < min_face: + skipped += 1 + continue count += 1 res = check_graph(G) if res is not None: @@ -180,8 +211,10 @@ def main(): break elapsed = time.time() - start if found is None: - print(f"n={n:>3}: checked {count} graphs, no counterexample " - f"[{elapsed:.1f}s]") + extra = (f", skipped {skipped} due to small face" + if min_face is not None else "") + print(f"n={n:>3}: checked {count} graphs, no counterexample" + f"{extra} [{elapsed:.1f}s]") else: G, (col, K0, K1, h0, h1, a, b, c, e) = found colour_name = {0: 'red', 1: 'blue', 2: 'green'} diff --git a/papers/face_monochromatic_pairs/experiments/verify_28_vertex_counterexample.py b/papers/face_monochromatic_pairs/experiments/verify_28_vertex_counterexample.py new file mode 100644 index 0000000..000b7d7 --- /dev/null +++ b/papers/face_monochromatic_pairs/experiments/verify_28_vertex_counterexample.py @@ -0,0 +1,120 @@ +"""Verify the n=28 counterexample (smallest dual of min-degree-5 +triangulation that violates the face-length-≥-5 form of Conjecture +5.5) and render a planar PNG of it. + +The graph H is the planar dual of a 16-vertex triangulation with +min degree ≥ 5 (the 3rd one in sage's enumeration). It has 28 +vertices, 42 edges, and all faces of length ≥ 5. The colouring shown +below makes K_{red, blue} and K_{red, green} both 12-cycles sharing +the colour-red edge (0, 1) with h_φ ≡ -1 on each. + +Run with: sage experiments/verify_28_vertex_counterexample.py +""" +import os +import sys +import math + +from sage.all import Graph + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +from search_smaller_counterexample import ( + edge_key, heawood_numbers, trace_kempe, +) + +OUT_PNG = os.path.join(HERE, '..', 'figures', 'min-face-5-counterexample.png') + +# Dual edges and the discovered colouring (sorted-edge order). +EDGES_LIST = [ + (0, 1), (0, 4), (0, 6), (1, 2), (1, 5), (2, 3), (2, 8), (3, 4), + (3, 11), (4, 13), (5, 7), (5, 9), (6, 7), (6, 15), (7, 17), + (8, 10), (8, 12), (9, 10), (9, 19), (10, 20), (11, 12), (11, 14), + (12, 22), (13, 14), (13, 16), (14, 23), (15, 16), (15, 18), + (16, 25), (17, 18), (17, 19), (18, 26), (19, 21), (20, 21), + (20, 22), (21, 27), (22, 24), (23, 24), (23, 25), (24, 27), + (25, 26), (26, 27), +] +COLOURING = (0, 1, 2, 2, 1, 1, 0, 2, 0, 0, 2, 0, 1, 0, 0, 1, 2, 2, + 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 0, 2, 1, 1, + 2, 1, 2, 0, 1, 2) +COLOUR_NAME = {0: 'red', 1: 'blue', 2: 'green'} + + +def build(): + G = Graph(multiedges=False, loops=False) + for u, v in EDGES_LIST: + G.add_edge(u, v) + # Sage's sorted edges should align with EDGES_LIST. + edges_sorted = sorted([edge_key(u, v) for (u, v) in G.edge_iterator(labels=False)]) + assert edges_sorted == EDGES_LIST, "edge order mismatch" + col_of_edge = {edges_sorted[i]: COLOURING[i] for i in range(len(edges_sorted))} + return G, col_of_edge + + +def main(): + G, col_of_edge = build() + print(f"|V| = {G.order()}, |E| = {G.size()}") + assert G.is_planar(set_embedding=True) + print(f"face lengths: {sorted([len(f) for f in G.faces()])}") + assert all(len(f) >= 5 for f in G.faces()), "face length < 5" + # Verify proper 3-edge-colouring. + for v in G.vertex_iterator(): + cs = sorted(col_of_edge[edge_key(v, u)] for u in G.neighbors(v)) + assert cs == [0, 1, 2], f"vertex {v} colours {cs}" + print("proper 3-edge-colouring ✓") + + h = heawood_numbers(G, col_of_edge) + plus = sum(1 for v in h if h[v] == +1) + minus = sum(1 for v in h if h[v] == -1) + print(f"global h_φ: {plus} (+1) / {minus} (-1)") + + # Verify both Kempe cycles through (0, 1) are constant. + K0 = trace_kempe(G, col_of_edge, (0, 1), (0, 1)) # red+blue + K1 = trace_kempe(G, col_of_edge, (0, 1), (0, 2)) # red+green + h_K0 = [h[v] for v in K0] + h_K1 = [h[v] for v in K1] + print(f"K_{{red, blue}} (len {len(K0)}): {K0}, h = {h_K0[0]} (const? {len(set(h_K0))==1})") + print(f"K_{{red, green}} (len {len(K1)}): {K1}, h = {h_K1[0]} (const? {len(set(h_K1))==1})") + print(f"V(K0) ∪ V(K1)| = {len(set(K0) | set(K1))} / {G.order()}") + + # Canonical graph6 + print(f"canonical graph6 = {G.canonical_label().graph6_string()}") + + # Render PNG with planar layout + import matplotlib + matplotlib.use('Agg') + import matplotlib.pyplot as plt + + pos = G.layout_planar() + fig, ax = plt.subplots(figsize=(10, 10), dpi=160) + ax.set_aspect('equal') + ax.axis('off') + for (u, v) in G.edge_iterator(labels=False): + c = COLOUR_NAME[col_of_edge[edge_key(u, v)]] + x1, y1 = pos[u]; x2, y2 = pos[v] + ax.plot([x1, x2], [y1, y2], color=c, linewidth=2.0, solid_capstyle='round') + K0set = set(K0); K1set = set(K1) + for v, (x, y) in pos.items(): + if v in K0set and v in K1set: + face_c = '#bbbbff' # on both cycles + elif v in K0set or v in K1set: + face_c = '#ddddff' + else: + face_c = 'lightgrey' + ax.plot(x, y, 'o', markersize=18, markerfacecolor=face_c, + markeredgecolor='black', markeredgewidth=1.0) + ax.text(x, y, str(v), ha='center', va='center', fontsize=8) + xs = [p[0] for p in pos.values()] + ys = [p[1] for p in pos.values()] + pad = 0.5 + ax.set_xlim(min(xs) - pad, max(xs) + pad) + ax.set_ylim(min(ys) - pad, max(ys) + pad) + os.makedirs(os.path.dirname(OUT_PNG), exist_ok=True) + fig.tight_layout() + fig.savefig(OUT_PNG, dpi=160, bbox_inches='tight') + plt.close(fig) + print(f"\nWrote: {OUT_PNG}") + + +if __name__ == '__main__': + main() diff --git a/papers/face_monochromatic_pairs/figures/min-face-5-counterexample.png b/papers/face_monochromatic_pairs/figures/min-face-5-counterexample.png new file mode 100644 index 0000000000000000000000000000000000000000..a17418674887a87bb7ecea54d8bd0a5e2aa1d801 GIT binary patch literal 192060 zcmdRWXH-*L7p+$mu^=iUh!h(jAc9osO{9ZTrGrTCT?h~m6s1a+F4Ak1-a@f}bcjIc z0V$#P1PFw@10?9ZpYO-}`7(y%>K)nH`|Q2;Tyw28*LnL`Nt*Nw^_gSGj*-gBJXAS$ zjF@Bp4+$~we}>J7_kjNhI7>ZsR<$#Ab~AD`Ii_gjY;SGnY;E!Ex~qw!lZBlv*S-53 z_jvAJH+OcncM@P{w>kR8Jv&D;cC8(ALEu+T+skM<9Xm#qu>a>DSr@;kWB(jGCi_ra z?M33k2hzZ?AI7+}Z~Fy1Mi@S1iZ5gQ=T?rYzr;g+?!d0V zl4oAWwV6J4aIw7gkGv%-|AOSw`DopDN@s6&KM5joCuY6$@NKy2CC4qjO?|qeqQ}<< zkXUa$O1M(38(N#s>g3*2gcpKdQ&m+p5?I;c=eUf29TVXn$GR?&o+JF@g1xyR;qN{^ zl0-y=f1IG9rXu|P`dC*z;qS-(2_W9zzk?6S18!hP4?kk{slLENAAXLX|Nm-=#mi&V z)YKd0by!8a+U?9yuT8xqA$JuQ7ni`$Pz_{963>ejLsD7+HKJ4J)t*03ugf}1cv6W{ 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