dual_decomposition: swap algorithm trace to n=14 + final-graph conjecture
- Replace the dodecahedron trace at the end of section 3 with the n=14
triangulation found by search_kempe_property.py: its H_1 admits a
proper 3-edge-colouring satisfying both chord-apex and Kempe-cycle
conditions (Lemmas 2.6, 2.7).
- experiments/draw_iterated_reduction_n14.py: rebuilds fig_alg_step{0,1,2}
with Tutte barycentric layouts (outer face chosen to keep v_n in the
interior); also runs the algorithm to completion, checking chord-apex +
Kempe at each step (step 1 satisfies all; step 2 fails chord-apex;
step 3 terminates).
- Add Conjecture 3.4: G is a minimal counterexample iff no proper
3-edge-colouring of the final reduced graph H_{t*} has all (spike_t,
merged_t) pairs in distinct colours.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
+520
@@ -0,0 +1,520 @@
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"""Draw the iterated reduction trace on the smallest triangulation where
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the chord-apex + Kempe-cycle property is satisfied: the first min-degree-5
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plantri triangulation on n = 14 vertices, found by search_kempe_property.py.
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Overwrites fig_alg_step{0,1,2}.png in the paper directory with this
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triangulation's trace (replacing the dodecahedron version).
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Run with: sage experiments/draw_iterated_reduction_n14.py
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"""
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from sage.all import Graph
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from sage.graphs.graph_generators import graphs
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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import math
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import os
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def tutte_layout(G_sage, avoid_verts=None, iterations=300):
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"""Tutte's barycentric embedding: pick the largest face whose vertex set
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avoids `avoid_verts` as the outer face, place its vertices on a regular
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polygon, then iterate each interior vertex to the barycenter of its
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neighbors. For 3-connected planar graphs this converges to the unique
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straight-line planar embedding with the chosen outer face --- balanced
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by construction and free of edge crossings."""
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avoid = set(avoid_verts or ())
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candidates = []
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for face in G_sage.faces():
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verts = [u for (u, v) in face]
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if not (set(verts) & avoid):
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candidates.append(verts)
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if not candidates:
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outer = [u for (u, v) in max(G_sage.faces(), key=len)]
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else:
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outer = max(candidates, key=len)
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n_outer = len(outer)
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pos = {}
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for k, v in enumerate(outer):
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ang = 2 * math.pi * k / n_outer + math.pi / 2
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pos[v] = (math.cos(ang), math.sin(ang))
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interior = [v for v in G_sage.vertex_iterator() if v not in pos]
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for v in interior:
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pos[v] = (0.0, 0.0)
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for _ in range(iterations):
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new_pos = dict(pos)
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for v in interior:
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nbrs = list(G_sage.neighbor_iterator(v))
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sx = sum(pos[w][0] for w in nbrs) / len(nbrs)
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sy = sum(pos[w][1] for w in nbrs) / len(nbrs)
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new_pos[v] = (sx, sy)
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pos = new_pos
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return pos
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OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
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C = ['#dc2626', '#16a34a', '#2563eb']
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GRAY = '#9ca3af'
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DARK = '#374151'
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HIGHLIGHT = '#fef3c7'
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SHADE = '#fef3c7'
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def dual_of(G):
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faces = G.faces()
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edge_to_faces = {}
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for fi, face in enumerate(faces):
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for u, v in face:
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e = frozenset((u, v))
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edge_to_faces.setdefault(e, []).append(fi)
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dual_edges = []
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for e, fs in edge_to_faces.items():
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if len(fs) == 2:
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dual_edges.append((fs[0], fs[1]))
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return Graph(dual_edges, multiedges=False, loops=False)
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def apply_reduction(G, face, i, v_n_label):
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boundary = [u for (u, v) in face]
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if len(set(boundary)) != 5:
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return None
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A = []
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for B_k in boundary:
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outer = [w for w in G.neighbor_iterator(B_k) if w not in boundary]
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if len(outer) != 1:
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return None
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A.append(outer[0])
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if len(set(A)) != 5:
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return None
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if A[(i + 3) % 5] == A[(i + 4) % 5]:
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return None
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H = G.copy()
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for v in boundary:
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H.delete_vertex(v)
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H.add_vertex(v_n_label)
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side_0 = (v_n_label, A[i % 5])
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spike = (v_n_label, A[(i + 1) % 5])
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side_1 = (v_n_label, A[(i + 2) % 5])
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merged = (A[(i + 3) % 5], A[(i + 4) % 5])
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H.add_edges([side_0, spike, side_1, merged])
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if H.has_multiple_edges() or H.has_loops():
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return None
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if not H.is_planar(set_embedding=True):
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return None
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if not all(H.degree(v) == 3 for v in H.vertex_iterator()):
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return None
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named = {
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'spike': frozenset(spike),
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'side_0': frozenset(side_0),
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'side_1': frozenset(side_1),
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'merged': frozenset(merged),
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}
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return H, named, boundary, A
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def proper_3_edge_colorings(G):
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edges = list(G.edges(labels=False))
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n_edges = len(edges)
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adj = [[] for _ in range(n_edges)]
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for i in range(n_edges):
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u, v = edges[i][0], edges[i][1]
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for j in range(i):
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x, y = edges[j][0], edges[j][1]
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if u in (x, y) or v in (x, y):
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adj[i].append(j)
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adj[j].append(i)
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coloring = [-1] * n_edges
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def back(k):
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if k == n_edges:
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yield tuple(coloring)
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return
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for c in range(3):
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if all(coloring[j] != c for j in adj[k]):
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coloring[k] = c
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yield from back(k + 1)
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coloring[k] = -1
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return edges, back(0)
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def kempe_cycle(edges, coloring, start_idx, color_pair):
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a, b = color_pair
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in_sub = [i for i in range(len(edges)) if coloring[i] in (a, b)]
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if start_idx not in in_sub:
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return None
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visited = {start_idx}
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stack = [start_idx]
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while stack:
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cur = stack.pop()
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u, v = edges[cur][0], edges[cur][1]
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for j in in_sub:
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if j in visited:
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continue
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x, y = edges[j][0], edges[j][1]
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if u in (x, y) or v in (x, y):
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visited.add(j)
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stack.append(j)
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return visited
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def matches_property(edges, col, named):
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idx = {}
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for ii, e in enumerate(edges):
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es = frozenset((e[0], e[1]))
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for role, ns in named.items():
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if es == ns:
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idx[role] = ii
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if len(idx) != 4:
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return False
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c_spike = col[idx['spike']]
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c_merged = col[idx['merged']]
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if c_spike != c_merged:
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return False
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c_s0 = col[idx['side_0']]
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c_s1 = col[idx['side_1']]
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kc0 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s0))
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if idx['side_0'] not in kc0 or idx['merged'] not in kc0:
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return False
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kc1 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s1))
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if idx['side_1'] not in kc1 or idx['merged'] not in kc1:
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return False
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return True
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def find_first_match():
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"""Iterate over (G, face, i_red, coloring) and return the first hit."""
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for G in graphs.triangulations(14, minimum_degree=5):
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if not G.is_planar(set_embedding=True):
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continue
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D = dual_of(G)
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D.is_planar(set_embedding=True)
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for face in D.faces():
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if len(face) != 5:
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continue
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for i_red in range(5):
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res = apply_reduction(D, face, i_red, '__v_n_1__')
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if res is None:
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continue
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H, named, boundary, A = res
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edges, gen = proper_3_edge_colorings(H)
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for col in gen:
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if matches_property(edges, col, named):
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coloring_dict = {frozenset((e[0], e[1])): c
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for e, c in zip(edges, col)}
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return G, D, face, i_red, H, named, boundary, A, coloring_dict
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return None
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def draw_graph(ax, G, pos, *, coloring=None, protected=None,
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shade_vertices=None, vn_labels=None):
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if shade_vertices:
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poly = [pos[v] for v in shade_vertices]
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ax.add_patch(Polygon(poly, closed=True, facecolor=SHADE,
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edgecolor='none', zorder=0))
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protected = protected or set()
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vn_labels = vn_labels or {}
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for u, v, _ in G.edges():
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e = frozenset([u, v])
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c = C[coloring[e]] if (coloring is not None and e in coloring) else GRAY
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lw = 3.8 if e in protected else 1.4
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(x0, y0), (x1, y1) = pos[u], pos[v]
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ax.plot([x0, x1], [y0, y1], color=c, lw=lw, zorder=2)
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for v in G.vertices(sort=False):
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x, y = pos[v]
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if v in vn_labels:
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ax.scatter(x, y, s=320, color=HIGHLIGHT, marker='s',
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edgecolors='black', linewidths=1.2, zorder=4)
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ax.annotate(vn_labels[v], (x, y),
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textcoords='offset points', xytext=(16, 16),
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ha='left', fontsize=14, fontweight='bold',
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color=DARK, zorder=6,
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bbox=dict(boxstyle='round,pad=0.2', fc='white',
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ec=DARK, lw=0.6))
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else:
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ax.scatter(x, y, s=60, color=DARK, zorder=3)
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ax.set_aspect('equal')
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ax.axis('off')
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def main():
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print("Searching for the first match at n = 14 ...")
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result = find_first_match()
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if result is None:
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print("No match found at n = 14.")
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return
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G14, D, face, i_red, H1, named1, boundary1, A1, coloring1 = result
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print(f"Found at i_red = {i_red}")
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print(f" G (n=14): |V|={G14.order()}, |E|={G14.size()}, "
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f"min_deg={min(G14.degree())}")
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print(f" D = G': |V|={D.order()}, |E|={D.size()}")
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print(f" H_1: |V|={H1.order()}, |E|={H1.size()}")
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# Relabel H_1 in place so all vertex labels are comparable integers
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# (Sage's planar layout and face enumeration need comparable labels).
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# Translate coloring1 and named1 accordingly.
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H1_relabel_map = {v: i for i, v in enumerate(H1.vertex_iterator())}
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H1.relabel(perm=H1_relabel_map, inplace=True)
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vn1_int = H1_relabel_map['__v_n_1__']
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coloring1 = {frozenset(H1_relabel_map[u] for u in e): c
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for e, c in coloring1.items()}
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named1 = {role: frozenset(H1_relabel_map[u] for u in e)
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for role, e in named1.items()}
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D.is_planar(set_embedding=True)
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D_layout = tutte_layout(D, avoid_verts=set(u for (u, v) in face))
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H1.is_planar(set_embedding=True)
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H1_layout = tutte_layout(H1, avoid_verts={vn1_int})
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boundary_face_verts = [u for (u, v) in face]
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fig, ax = plt.subplots(figsize=(8, 8))
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draw_graph(ax, D, D_layout, shade_vertices=boundary_face_verts)
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fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step0.png'),
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dpi=170, bbox_inches='tight')
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plt.close(fig)
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print("Wrote fig_alg_step0.png")
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E = set(named1.values())
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fig, ax = plt.subplots(figsize=(8, 8))
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draw_graph(ax, H1, H1_layout, coloring=coloring1, protected=E,
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vn_labels={vn1_int: '$v_n^{(1)}$'})
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fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step1.png'),
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dpi=170, bbox_inches='tight')
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plt.close(fig)
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print("Wrote fig_alg_step1.png")
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# ----- Step 2: try to continue -----
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H1.is_planar(set_embedding=True)
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chosen2 = None
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for face2 in H1.faces():
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if len(face2) != 5:
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continue
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boundary2 = [u for (u, v) in face2]
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boundary2_edges = [frozenset([u, v]) for (u, v) in face2]
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externals2 = []
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A2 = []
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valid_face = True
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for B_k in boundary2:
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outer = [w for w in H1.neighbor_iterator(B_k) if w not in boundary2]
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if len(outer) != 1:
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valid_face = False
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break
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externals2.append(frozenset([B_k, outer[0]]))
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A2.append(outer[0])
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if not valid_face:
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continue
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if any(e in E for e in boundary2_edges + externals2):
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continue
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# find valid i_t
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f_vec = [coloring1[e] for e in externals2]
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for i_t in range(5):
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if f_vec[(i_t + 3) % 5] != f_vec[(i_t + 4) % 5]:
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continue
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if len({f_vec[i_t], f_vec[(i_t + 1) % 5], f_vec[(i_t + 2) % 5]}) != 3:
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continue
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if A2[(i_t + 3) % 5] == A2[(i_t + 4) % 5]:
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continue
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chosen2 = (face2, i_t, boundary2, externals2, A2)
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break
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if chosen2 is not None:
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break
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if chosen2 is None:
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# algorithm terminates at H_1
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fig, ax = plt.subplots(figsize=(8, 8))
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ax.text(0.5, 0.5,
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"Algorithm terminates at $H_1$:\n"
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"no pentagonal face of $H_1$ has all\n"
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"ten incident edges outside $E$.",
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ha='center', va='center', fontsize=18, color=DARK,
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transform=ax.transAxes,
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bbox=dict(boxstyle='round,pad=0.6', fc=HIGHLIGHT,
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ec=DARK, lw=1.0))
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ax.set_aspect('equal')
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ax.axis('off')
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fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step2.png'),
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dpi=170, bbox_inches='tight')
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plt.close(fig)
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print("Wrote fig_alg_step2.png (termination card)")
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print(" Algorithm terminates at H_1: no safe pentagonal face.")
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return
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face2, i_t, boundary2, externals2, A2 = chosen2
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print(f"Step 2: safe face found, i_t = {i_t}")
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H2 = H1.copy()
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for v in boundary2:
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H2.delete_vertex(v)
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# use a fresh int label for v_n^(2)
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v_n_2 = max(H1.vertices(sort=False)) + 1
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H2.add_vertex(v_n_2)
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side_0_2 = (v_n_2, A2[i_t])
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spike_2 = (v_n_2, A2[(i_t + 1) % 5])
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side_1_2 = (v_n_2, A2[(i_t + 2) % 5])
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merged_2 = (A2[(i_t + 3) % 5], A2[(i_t + 4) % 5])
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H2.add_edges([side_0_2, spike_2, side_1_2, merged_2])
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H2.is_planar(set_embedding=True)
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|
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coloring2 = {e: c for e, c in coloring1.items()
|
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if not any(u in boundary2 for u in e)}
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coloring2[frozenset(side_0_2)] = coloring1[externals2[i_t]]
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coloring2[frozenset(spike_2)] = coloring1[externals2[(i_t + 1) % 5]]
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coloring2[frozenset(side_1_2)] = coloring1[externals2[(i_t + 2) % 5]]
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coloring2[frozenset(merged_2)] = coloring1[externals2[(i_t + 3) % 5]]
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E |= {frozenset(side_0_2), frozenset(spike_2),
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frozenset(side_1_2), frozenset(merged_2)}
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H2_layout = tutte_layout(H2, avoid_verts={vn1_int, v_n_2})
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fig, ax = plt.subplots(figsize=(8, 8))
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draw_graph(ax, H2, H2_layout, coloring=coloring2, protected=E,
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vn_labels={vn1_int: '$v_n^{(1)}$',
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v_n_2: '$v_n^{(2)}$'})
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fig.savefig(os.path.join(OUT_DIR, 'fig_alg_step2.png'),
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dpi=170, bbox_inches='tight')
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plt.close(fig)
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print(f"Wrote fig_alg_step2.png: H_2 with |V|={H2.order()}, "
|
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f"|E|={H2.size()}, |protected|={len(E)}")
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# --- continue running to completion, checking Kempe condition each step --
|
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print()
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print("=" * 72)
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print("Running algorithm to completion, checking chord-apex + Kempe at "
|
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"each step.")
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print("=" * 72)
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# Step 1 status (by construction this is the matching coloring)
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cond1 = check_step_conditions(H1, coloring1, named1)
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print(f" step t = 1: |V|={H1.order():>3}, |E_graph|={H1.size():>3}, "
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f"|E_prot|= 4 (initial)"
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f" | chord-apex: {cond1['chord_apex']}, "
|
||||
f"side_0-Kempe: {cond1['kc_side_0']}, "
|
||||
f"side_1-Kempe: {cond1['kc_side_1']}")
|
||||
run_to_completion_from(H2, coloring2, E,
|
||||
{'spike': frozenset(spike_2),
|
||||
'side_0': frozenset(side_0_2),
|
||||
'side_1': frozenset(side_1_2),
|
||||
'merged': frozenset(merged_2)},
|
||||
start_t=2)
|
||||
|
||||
|
||||
def check_step_conditions(H, coloring, named):
|
||||
"""Given an H_t and the *just-added* spike/side_0/side_1/merged, check
|
||||
whether chord-apex and the two Kempe-cycle conditions hold."""
|
||||
edges = list(H.edges(labels=False))
|
||||
edges_fs = [frozenset((u, v)) for (u, v) in edges]
|
||||
col = [coloring[e] for e in edges_fs]
|
||||
idx = {role: edges_fs.index(e) for role, e in named.items()}
|
||||
c_spike = col[idx['spike']]
|
||||
c_merged = col[idx['merged']]
|
||||
chord_apex = (c_spike == c_merged)
|
||||
if not chord_apex:
|
||||
return {'chord_apex': False, 'kc_side_0': False, 'kc_side_1': False}
|
||||
c_s0 = col[idx['side_0']]
|
||||
c_s1 = col[idx['side_1']]
|
||||
kc0 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s0))
|
||||
kc1 = kempe_cycle(edges, col, idx['spike'], (c_spike, c_s1))
|
||||
kc_side_0 = (idx['side_0'] in kc0 and idx['merged'] in kc0)
|
||||
kc_side_1 = (idx['side_1'] in kc1 and idx['merged'] in kc1)
|
||||
return {'chord_apex': True, 'kc_side_0': kc_side_0, 'kc_side_1': kc_side_1}
|
||||
|
||||
|
||||
def find_safe_face(H, protected):
|
||||
"""Return (face, externals, A) for some safe pentagonal face avoiding
|
||||
`protected`, or None."""
|
||||
for face in H.faces():
|
||||
if len(face) != 5:
|
||||
continue
|
||||
boundary = [u for (u, v) in face]
|
||||
boundary_edges = [frozenset([u, v]) for (u, v) in face]
|
||||
externals = []
|
||||
A = []
|
||||
valid = True
|
||||
for B_k in boundary:
|
||||
outer = [w for w in H.neighbor_iterator(B_k) if w not in boundary]
|
||||
if len(outer) != 1:
|
||||
valid = False
|
||||
break
|
||||
externals.append(frozenset([B_k, outer[0]]))
|
||||
A.append(outer[0])
|
||||
if not valid:
|
||||
continue
|
||||
if any(e in protected for e in boundary_edges + externals):
|
||||
continue
|
||||
return face, boundary, externals, A
|
||||
return None
|
||||
|
||||
|
||||
def run_to_completion_from(H, coloring, E, last_named, start_t):
|
||||
"""Continue iterating from H_{start_t}. The 'last_named' dict carries
|
||||
the spike/side/merged of step `start_t` so we can report its Kempe
|
||||
status. Print a row per step."""
|
||||
t = start_t
|
||||
print(f" step t = {t}: |V|={H.order():>3}, |E_graph|={H.size():>3}, "
|
||||
f"|E_prot|={len(E):>3}", end='')
|
||||
cond = check_step_conditions(H, coloring, last_named)
|
||||
print(f" | chord-apex: {cond['chord_apex']}, "
|
||||
f"side_0-Kempe: {cond['kc_side_0']}, "
|
||||
f"side_1-Kempe: {cond['kc_side_1']}")
|
||||
|
||||
while True:
|
||||
H.is_planar(set_embedding=True)
|
||||
res = find_safe_face(H, E)
|
||||
if res is None:
|
||||
print(f" step t = {t + 1}: no safe pentagonal face --> "
|
||||
f"algorithm terminates at H_{t}.")
|
||||
return
|
||||
face, boundary, externals, A = res
|
||||
f_vec = [coloring[e] for e in externals]
|
||||
i_t = None
|
||||
for i in range(5):
|
||||
if f_vec[(i + 3) % 5] != f_vec[(i + 4) % 5]:
|
||||
continue
|
||||
if len({f_vec[i], f_vec[(i + 1) % 5], f_vec[(i + 2) % 5]}) != 3:
|
||||
continue
|
||||
if A[(i + 3) % 5] == A[(i + 4) % 5]:
|
||||
continue
|
||||
i_t = i
|
||||
break
|
||||
if i_t is None:
|
||||
print(f" step t = {t + 1}: f = {f_vec}, no valid index --> "
|
||||
f"terminate (Lemma 2.4 violation? Probably a parallel-edge "
|
||||
f"or other degenerate case).")
|
||||
return
|
||||
t += 1
|
||||
v_n_new = max(H.vertices(sort=False)) + 1 if all(
|
||||
isinstance(v, int) for v in H.vertex_iterator()) else f'vn{t}'
|
||||
H_new = H.copy()
|
||||
for v in boundary:
|
||||
H_new.delete_vertex(v)
|
||||
H_new.add_vertex(v_n_new)
|
||||
side_0 = (v_n_new, A[i_t])
|
||||
spike = (v_n_new, A[(i_t + 1) % 5])
|
||||
side_1 = (v_n_new, A[(i_t + 2) % 5])
|
||||
merged = (A[(i_t + 3) % 5], A[(i_t + 4) % 5])
|
||||
H_new.add_edges([side_0, spike, side_1, merged])
|
||||
H = H_new
|
||||
coloring = {e: c for e, c in coloring.items()
|
||||
if not any(u in boundary for u in e)}
|
||||
coloring[frozenset(side_0)] = coloring[externals[i_t]] \
|
||||
if frozenset(externals[i_t]) in coloring else f_vec[i_t]
|
||||
# safer: directly use f_vec
|
||||
coloring[frozenset(side_0)] = f_vec[i_t]
|
||||
coloring[frozenset(spike)] = f_vec[(i_t + 1) % 5]
|
||||
coloring[frozenset(side_1)] = f_vec[(i_t + 2) % 5]
|
||||
coloring[frozenset(merged)] = f_vec[(i_t + 3) % 5]
|
||||
named = {
|
||||
'spike': frozenset(spike),
|
||||
'side_0': frozenset(side_0),
|
||||
'side_1': frozenset(side_1),
|
||||
'merged': frozenset(merged),
|
||||
}
|
||||
E |= set(named.values())
|
||||
cond = check_step_conditions(H, coloring, named)
|
||||
print(f" step t = {t}: |V|={H.order():>3}, |E_graph|={H.size():>3}, "
|
||||
f"|E_prot|={len(E):>3}, i_t = {i_t}", end='')
|
||||
print(f" | chord-apex: {cond['chord_apex']}, "
|
||||
f"side_0-Kempe: {cond['kc_side_0']}, "
|
||||
f"side_1-Kempe: {cond['kc_side_1']}")
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
main()
|
||||
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@@ -22,6 +22,7 @@
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Algorithm\nonbreakingspace 3.1\hbox {} on $G' = $ dodecahedron (dual of the icosahedron). \emph {Left:} $G'$ (20 vertices, 30 edges), with $F_v$ (the inner pentagon) shaded as the face chosen for the first reduction. \emph {Centre:} $H_1$ (16 vertices, 24 edges) after step\nonbreakingspace (1) with $i_1 = 0$, $3$-edge-coloured by Sage; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker. \emph {Right:} $H_2$ (12 vertices, 18 edges) after step\nonbreakingspace (3) with $i_t = 0$; the only safe pentagonal face in $H_1$ was the outer pentagon, whose deletion produces $v_n^{(2)}$ and a second chord, giving eight protected edges. No safe pentagonal face remains, so the algorithm terminates. The generating script is \texttt {experiments/draw\_iterated\_reduction.py}.}}{8}{}\protected@file@percent }
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Algorithm\nonbreakingspace 3.1\hbox {} on $G'=\mathrm {dual}(G)$, where $G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and $\varphi _1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies both the chord-apex condition (Lemma\nonbreakingspace 2.6\hbox {}) and the Kempe-cycle condition (Lemma\nonbreakingspace 2.7\hbox {}), found by \texttt {experiments/search\_kempe\_property.py}. \emph {Left:} $G'$ ($24$ vertices, $36$ edges) with the chosen pentagonal face shaded. \emph {Centre:} $H_1$ ($20$ vertices, $30$ edges) after step\nonbreakingspace (1) with $i_1 = 1$, $3$-edge-coloured by $\varphi _1$; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share the colour green. \emph {Right:} $H_2$ ($16$ vertices, $24$ edges) after step\nonbreakingspace (3) with $i_t = 3$; eight edges are protected, and the algorithm terminates one step later (no remaining safe pentagonal face in $H_2$). The generating script is \texttt {experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte barycentric embeddings with the outer face picked to keep $v_n^{(1)}, v_n^{(2)}$ in the interior.}}{8}{}\protected@file@percent }
|
||||
\newlabel{fig:iterated-reduction-trace}{{3}{8}}
|
||||
\newlabel{conj:no-all-distinct-coloring}{{3.4}{8}}
|
||||
\gdef \@abspage@last{8}
|
||||
|
||||
@@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 23 MAY 2026 03:21
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 23 MAY 2026 13:19
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
%&-line parsing enabled.
|
||||
@@ -144,26 +144,26 @@ File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
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||||
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|
||||
\openout1 = `paper.aux'.
|
||||
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@@ -192,7 +192,7 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
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e
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[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
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||||
[]\OT1/cmr/m/n/10 List the five degree-$2$ ver-tices in clock-wise or-der aroun
|
||||
d $\OML/cmm/m/it/10 F$ \OT1/cmr/m/n/10 as $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 =
|
||||
(\OML/cmm/m/it/10 A[]; A[]; A[]; A[]; A[]\OT1/cmr/m/n/10 )$.
|
||||
@@ -201,22 +201,22 @@ d $\OML/cmm/m/it/10 F$ \OT1/cmr/m/n/10 as $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 =
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<fig_reduced_dual_step1.png, id=17, 517.79329pt x 499.08812pt>
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[4] [5 <./fig_chord_apex_step1.png> <./fig_chord_apex_step2.png> <./fig_chord_a
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||||
pex_step3.png>] [6]
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\OT1/cmr/m/n/10 which $\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[
|
||||
]\OT1/cmr/m/n/10 ) = \OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\
|
||||
OT1/cmr/m/n/10 )$ and $\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[
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@@ -253,31 +253,60 @@ T1/cmr/m/n/10 )\OML/cmm/m/it/10 ; '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\OT1/
|
||||
cmr/m/n/10 )$
|
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[]
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\OT1/cmr/m/sc/10 Figure 3.\OT1/cmr/m/n/10 Algorithm 3.1[] on $\OML/cmm/m/it/10
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G[] \OT1/cmr/m/n/10 = [](\OML/cmm/m/it/10 G\OT1/cmr/m/n/10 )$, where $\OML/cmm/
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m/it/10 G$
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[]
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Underfull \hbox (badness 3623) in paragraph at lines 498--498
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\OT1/cmr/m/n/10 is the first min-degree-$5$ plantri tri-an-gu-la-tion on $14$ v
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er-
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[]
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Underfull \hbox (badness 3179) in paragraph at lines 498--498
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\OT1/cmr/m/n/10 tices and $\OML/cmm/m/it/10 '[]$ \OT1/cmr/m/n/10 is a spe-cific
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[]
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\OT1/cmr/m/n/10 that sat-is-fies both the chord-apex con-di-tion (Lemma 2.6[])
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[]
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Underfull \hbox (badness 6094) in paragraph at lines 498--498
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\OT1/cmr/m/n/10 and the Kempe-cycle con-di-tion (Lemma 2.7[]), found by
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[]
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[7] [8 <./fig_alg_step0.png> <./fig_alg_step1.png> <./fig_alg_step2.png>]
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(./paper.aux) )
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\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{corollary}[theorem]{Corollary}
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\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{conjecture}[theorem]{Conjecture}
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\theoremstyle{definition}
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\newtheorem{definition}[theorem]{Definition}
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@@ -478,17 +479,36 @@ any further structure to $\varphi_t$ for $t \geq 2$ is left open.
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\includegraphics[width=0.32\textwidth]{fig_alg_step0.png}\hfill
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\includegraphics[width=0.32\textwidth]{fig_alg_step1.png}\hfill
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\includegraphics[width=0.32\textwidth]{fig_alg_step2.png}
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\caption{Algorithm~\ref{alg:iterated-reduction} on $G' = $ dodecahedron
|
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(dual of the icosahedron). \emph{Left:} $G'$ (20 vertices, 30 edges), with
|
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$F_v$ (the inner pentagon) shaded as the face chosen for the first reduction.
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\emph{Centre:} $H_1$ (16 vertices, 24 edges) after step~(1) with $i_1 = 0$,
|
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$3$-edge-coloured by Sage; the four edges around $v_n^{(1)}$ in $E$ are drawn
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thicker. \emph{Right:} $H_2$ (12 vertices, 18 edges) after step~(3) with
|
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$i_t = 0$; the only safe pentagonal face in $H_1$ was the outer pentagon,
|
||||
whose deletion produces $v_n^{(2)}$ and a second chord, giving eight protected
|
||||
edges. No safe pentagonal face remains, so the algorithm terminates. The
|
||||
generating script is \texttt{experiments/draw\_iterated\_reduction.py}.}
|
||||
\caption{Algorithm~\ref{alg:iterated-reduction} on $G'=\mathrm{dual}(G)$, where
|
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$G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and
|
||||
$\varphi_1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies
|
||||
both the chord-apex condition (Lemma~\ref{lem:chord-apex}) and the Kempe-cycle
|
||||
condition (Lemma~\ref{lem:kempe-spike}), found by
|
||||
\texttt{experiments/search\_kempe\_property.py}. \emph{Left:} $G'$
|
||||
($24$ vertices, $36$ edges) with the chosen pentagonal face shaded.
|
||||
\emph{Centre:} $H_1$ ($20$ vertices, $30$ edges) after step~(1) with
|
||||
$i_1 = 1$, $3$-edge-coloured by $\varphi_1$; the four edges around
|
||||
$v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share
|
||||
the colour green. \emph{Right:} $H_2$ ($16$ vertices, $24$ edges) after
|
||||
step~(3) with $i_t = 3$; eight edges are protected, and the algorithm
|
||||
terminates one step later (no remaining safe pentagonal face in $H_2$).
|
||||
The generating script is
|
||||
\texttt{experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte
|
||||
barycentric embeddings with the outer face picked to keep $v_n^{(1)},
|
||||
v_n^{(2)}$ in the interior.}
|
||||
\label{fig:iterated-reduction-trace}
|
||||
\end{figure}
|
||||
|
||||
\begin{conjecture}
|
||||
\label{conj:no-all-distinct-coloring}
|
||||
Let $G$ be a maximal planar graph of minimum degree $\geq 5$, and let
|
||||
$H_{t^*}$ be the final reduced graph produced by some terminating execution
|
||||
of Algorithm~\ref{alg:iterated-reduction} on $G$, with the corresponding
|
||||
sequence of named pairs $(\mathrm{spike}_t, \mathrm{merged}_t)$ for
|
||||
$t = 1, \dots, t^*$. Then $G$ is a minimal counterexample to the Four Colour
|
||||
Theorem if and only if there is no proper $3$-edge-colouring of $H_{t^*}$
|
||||
under which $\mathrm{spike}_t$ and $\mathrm{merged}_t$ receive different
|
||||
colours for every $t \in \{1, \dots, t^*\}$.
|
||||
\end{conjecture}
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user