didericis
9e86122818
Extends paper with: a notation section for color-class preimages; the plane diamond coloring definition (4-coloring whose two classes lift to a 2-coloring of some BFS-rooted diamond scaffold); a connectedness lemma for the scaffold; a proposition reformulating the property as parity- separation of two color classes by BFS layers; a remark noting this is strictly stronger than 4CT; and the conjecture that every maximal planar graph admits such a coloring. Adds plane_diamond_coloring.py with get_plane_diamond_scaffold and a counterexample search that reduces the per-root check to 4-colorability of an auxiliary graph forcing two colors onto opposite parity layers. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
75 lines
2.9 KiB
Python
75 lines
2.9 KiB
Python
"""Plane diamond coloring on maximal planar graphs."""
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from typing import Any
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from sage.all import Graph, graphs # type: ignore[attr-defined] # pylint: disable=no-name-in-module
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from lib.colored_graphs import canonize_and_save_graph
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def get_plane_diamond_scaffold(g: Graph, v: Any) -> Graph:
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"""
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Return the diamond scaffold of g relative to v.
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The diamond scaffold is the spanning subgraph of g obtained by removing
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every edge whose endpoints lie in the same level of the distance
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partition of g from v.
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"""
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distances = dict(g.breadth_first_search(v, report_distance=True))
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scaffold = g.copy()
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scaffold.delete_edges([(x, y) for x, y in g.edges(labels=False) if distances[x] == distances[y]])
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return scaffold
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def has_plane_diamond_coloring_at_root(g: Graph, u: Any) -> bool:
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"""
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Return True iff g admits a proper 4-coloring with two color classes
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parity-separated by the BFS distance partition from u.
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Equivalent to 4-colorability of the auxiliary graph H_u obtained by
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adjoining vertices v_a, v_b to g, with v_a adjacent to every odd-parity
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layer vertex, v_b adjacent to every even-parity layer vertex, and a
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v_a v_b edge.
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"""
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distances = dict(g.breadth_first_search(u, report_distance=True))
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odd_vertices = [v for v in g.vertices() if distances[v] % 2 == 1]
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even_vertices = [v for v in g.vertices() if distances[v] % 2 == 0]
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h = g.copy()
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v_a = max(g.vertices()) + 1
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v_b = v_a + 1
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h.add_vertex(v_a)
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h.add_vertex(v_b)
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h.add_edges([(v_a, w) for w in odd_vertices])
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h.add_edges([(v_b, w) for w in even_vertices])
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h.add_edge(v_a, v_b)
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return h.chromatic_number() <= 4
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def has_plane_diamond_coloring(g: Graph) -> bool:
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"""Return True iff some root vertex of g witnesses a plane diamond coloring."""
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return any(has_plane_diamond_coloring_at_root(g, u) for u in g.vertices())
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def search_counterexample(n: int, num_trials: int) -> Graph | None:
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"""
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Sample random maximal planar graphs of order n and return the first one
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with no plane diamond coloring, or None if none is found.
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"""
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for trial in range(num_trials):
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g = graphs.RandomTriangulation(n)
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if not has_plane_diamond_coloring(g):
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print(f"Counterexample found on trial {trial + 1}")
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return g
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if (trial + 1) % 10 == 0:
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print(f"Checked {trial + 1}/{num_trials} graphs of order {n}, no counterexample yet")
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return None
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if __name__ == "__main__":
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import sys
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n = int(sys.argv[1]) if len(sys.argv) > 1 else 12
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num_trials = int(sys.argv[2]) if len(sys.argv) > 2 else 100
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counterexample = search_counterexample(n, num_trials)
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if counterexample is None:
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print(f"No counterexample found in {num_trials} random triangulations of order {n}")
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else:
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canonical, graph_dir = canonize_and_save_graph(counterexample)
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print(f"Counterexample saved to {graph_dir}")
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