Add tire inner-boundary three-colour conjecture
Introduce Conjecture 1.31 (tire inner-boundary three-colour) as a decomposition-native weakening of the level-cycle conjecture 1.29: every maximal planar graph admits a vertex source and proper 4-colouring under which each tire inner boundary omits a colour. Remark 1.32 shows inner boundaries are single-level cycles, so the vertex-source form of 1.29 implies it on 2-connected boundaries. Extend check_level_cycle_three_color.py with --restriction inner-boundary (reconstructs the tire-tree decomposition from the embedding; inner boundary = level-(d+1) vertices of each depth-d dual component) and a --min-connectivity flag for the 5-connected slice. Verified: full census 4<=n<=13 (57716 triangulations) and 5-connected slice 14<=n<=24 (9732 graphs) all admit witnesses; no counterexample. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
@@ -1,24 +1,34 @@
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"""Empirical test for the level-cycle three-colour conjecture.
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"""Empirical test for the level-cycle three-colour conjecture and its
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tire inner-boundary refinement.
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The weakened conjecture says: for every maximal planar graph G, there
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is some level source S and some proper 4-vertex-colouring c such that
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every simple cycle contained in a single level G[L_d] uses at most
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three colours.
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The level-cycle conjecture (``--restriction level-cycle``, the default)
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says: for every maximal planar graph G, there is some level source S and
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some proper 4-vertex-colouring c such that every simple cycle contained
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in a single level G[L_d] uses at most three colours.
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Important: this checks the cycle-by-cycle version. Two cycles in the
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same level or the same inner outerplanar component may omit different
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colours.
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The inner-boundary conjecture (``--restriction inner-boundary``) is the
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weakening that constrains only the inner boundaries B_in of the tires in
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a tire-tree decomposition rooted at a vertex source: for every G there is
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a vertex source v0 and a proper 4-colouring c such that every tire inner
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boundary uses at most three colours. Because the inner outerplanar graph
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O of a depth-d tire satisfies O = G[component faces] cap L_{d+1} and is
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outerplanar, V(B_in) is exactly the level-(d+1) vertices of the tire's
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dual component (see the inner-boundary remark in the paper).
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Important: both checks are cycle-by-cycle. Two cycles in the same level
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or the same inner outerplanar component may omit different colours.
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Run examples:
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sage -python experiments/check_level_cycle_three_color.py 4 9
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sage -python experiments/check_level_cycle_three_color.py 4 8 --quantifier all-sources
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sage -python experiments/check_level_cycle_three_color.py 4 8 --sources all
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sage -python experiments/check_level_cycle_three_color.py 4 9 --restriction inner-boundary
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"""
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from __future__ import annotations
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import argparse
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from collections import deque
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from collections import defaultdict, deque
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from itertools import combinations
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from typing import Any, Iterable, Iterator, Sequence, cast
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@@ -131,10 +141,93 @@ def level_cycle_violation(
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return None
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def inner_boundary_vertex_sets(g: Graph, source: Source) -> list[frozenset[Any]]:
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"""Inner-boundary vertex sets of the tire-tree decomposition at a vertex.
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The tire-tree decomposition is defined only for a single-vertex source
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placed on the outer face. Each tire is a connected component of the
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depth-d dual subgraph G'_d (faces whose minimum vertex level is d). Because
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a depth-d face has its three vertex levels in {d, d+1} (adjacent vertices
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differ by at most one level), and because the inner outerplanar graph O of
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the tire is outerplanar (so every vertex of O lies on its inner-boundary
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walk), the inner-boundary vertex set equals the level-(d+1) vertices of the
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component: V(B_in) = V(O) = V(component) cap L_{d+1}.
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We work on the sphere dual (all faces). The choice of outer face does not
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change any inner-boundary set: depth->=1 components are untouched by removing
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a depth-0 face, and the root tire always has O = G[L_1].
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Only sets of size >= 4 are returned; smaller sets can never use four colours.
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"""
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if len(source) != 1:
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raise ValueError("inner-boundary restriction requires a vertex source")
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distances = distances_from_source(g, source)
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emb = cast(Graph, g.copy())
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if not emb.is_planar(set_embedding=True):
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raise ValueError("graph is not planar; cannot build tire decomposition")
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faces = emb.faces()
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face_vertices: list[set[Any]] = []
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for face in faces:
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verts: set[Any] = set()
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for edge in face:
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verts.add(edge[0])
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verts.add(edge[1])
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face_vertices.append(verts)
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depths = [min(distances[v] for v in verts) for verts in face_vertices]
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edge_faces: dict[frozenset[Any], list[int]] = defaultdict(list)
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for i, face in enumerate(faces):
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for edge in face:
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edge_faces[frozenset((edge[0], edge[1]))].append(i)
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dual_adj: dict[int, set[int]] = defaultdict(set)
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for incident in edge_faces.values():
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for a in range(len(incident)):
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for b in range(a + 1, len(incident)):
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dual_adj[incident[a]].add(incident[b])
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dual_adj[incident[b]].add(incident[a])
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targets: list[frozenset[Any]] = []
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seen = [False] * len(faces)
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for start in range(len(faces)):
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if seen[start]:
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continue
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depth = depths[start]
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component = [start]
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seen[start] = True
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stack = [start]
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while stack:
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f = stack.pop()
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for h in dual_adj[f]:
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if not seen[h] and depths[h] == depth:
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seen[h] = True
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component.append(h)
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stack.append(h)
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inner: set[Any] = set()
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for f in component:
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inner.update(v for v in face_vertices[f] if distances[v] == depth + 1)
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if len(inner) >= 4:
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targets.append(frozenset(inner))
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return targets
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def first_inner_boundary_violation(
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targets: Sequence[frozenset[Any]], coloring: Coloring
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) -> tuple[frozenset[Any], set[int]] | None:
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"""Return the first inner boundary using all four colours, if any."""
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for inner in targets:
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used = {coloring[v] for v in inner}
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if len(used) > 3:
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return inner, used
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return None
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def coloring_witness(
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g: Graph,
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source: Source,
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max_colorings: int | None,
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restriction: str,
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) -> tuple[Coloring | None, int, bool]:
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"""Find a proper 4-colouring satisfying the conjectured restriction.
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@@ -142,11 +235,20 @@ def coloring_witness(
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exhausted is False, max_colorings was reached before a decision.
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"""
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distances = distances_from_source(g, source)
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targets = (
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inner_boundary_vertex_sets(g, source)
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if restriction == "inner-boundary"
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else None
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)
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checked = 0
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for raw in all_graph_colorings(g, 4, vertex_color_dict=True):
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coloring = cast(Coloring, raw)
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checked += 1
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if level_cycle_violation(g, distances, coloring) is None:
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if restriction == "inner-boundary":
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violated = first_inner_boundary_violation(cast(Sequence, targets), coloring)
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else:
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violated = level_cycle_violation(g, distances, coloring)
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if violated is None:
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return coloring, checked, True
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if max_colorings is not None and checked >= max_colorings:
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return None, checked, False
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@@ -165,6 +267,7 @@ def test_graph(
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max_colorings: int | None,
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stop_first: bool,
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quantifier: str,
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restriction: str,
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) -> tuple[bool, bool, int]:
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"""Test a graph over selected sources.
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@@ -175,7 +278,9 @@ def test_graph(
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found_any_source = False
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for source in sources:
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checked_sources += 1
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witness, n_checked, exhausted = coloring_witness(g, source, max_colorings)
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witness, n_checked, exhausted = coloring_witness(
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g, source, max_colorings, restriction
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)
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if witness is None:
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if not exhausted:
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complete = False
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@@ -185,11 +290,16 @@ def test_graph(
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f"colorings_checked={n_checked}"
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)
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if exhausted and quantifier == "all-sources":
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distances = distances_from_source(g, source)
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first = next(all_graph_colorings(g, 4, vertex_color_dict=True), None)
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if first is not None:
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violation = level_cycle_violation(g, distances, cast(Coloring, first))
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print(f" first_coloring_violation={violation}")
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if restriction == "level-cycle":
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distances = distances_from_source(g, source)
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first = next(
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all_graph_colorings(g, 4, vertex_color_dict=True), None
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)
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if first is not None:
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violation = level_cycle_violation(
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g, distances, cast(Coloring, first)
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)
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print(f" first_coloring_violation={violation}")
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return False, complete, checked_sources
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if stop_first:
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if quantifier == "all-sources":
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@@ -232,12 +342,31 @@ def parse_args() -> argparse.Namespace:
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"the stronger earlier version"
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),
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)
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parser.add_argument(
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"--restriction",
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choices=("level-cycle", "inner-boundary"),
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default="level-cycle",
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help=(
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"level-cycle constrains every simple level cycle; inner-boundary "
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"constrains only the tire inner boundaries of a vertex-rooted "
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"tire-tree decomposition"
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),
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)
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parser.add_argument(
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"--max-cycle-source-size",
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type=int,
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default=None,
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help="optional cap on induced cycle source size",
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)
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parser.add_argument(
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"--min-connectivity",
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type=int,
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default=None,
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help=(
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"restrict to triangulations of at least this vertex connectivity "
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"(e.g. 5 for the 5-connected slice); passed to plantri via Sage"
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),
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)
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parser.add_argument(
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"--max-colorings",
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type=int,
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@@ -255,20 +384,37 @@ def parse_args() -> argparse.Namespace:
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def main() -> int:
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args = parse_args()
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stop_first = not args.full
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if args.restriction == "inner-boundary" and args.sources != "vertex":
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print(
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"note: inner-boundary restriction is vertex-rooted; "
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f"overriding --sources {args.sources} with vertex"
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)
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args.sources = "vertex"
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total_graphs = 0
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total_sources = 0
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unknown = 0
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triangulation_kwargs = {}
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if args.min_connectivity is not None:
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triangulation_kwargs["minimum_connectivity"] = args.min_connectivity
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for n in range(args.n_min, args.n_max + 1):
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print(f"=== n={n} ===")
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for idx, g in enumerate(graphs.triangulations(n), start=1):
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for idx, g in enumerate(
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graphs.triangulations(n, **triangulation_kwargs), start=1
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):
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total_graphs += 1
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source_list = list(
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level_sources(g, args.sources, args.max_cycle_source_size)
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)
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print(f" graph #{idx}: sources={len(source_list)}")
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passed, complete, checked_sources = test_graph(
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g, source_list, args.max_colorings, stop_first, args.quantifier
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g,
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source_list,
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args.max_colorings,
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stop_first,
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args.quantifier,
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args.restriction,
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)
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total_sources += checked_sources
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if not complete:
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@@ -276,7 +422,8 @@ def main() -> int:
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if not passed:
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print(
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f"COUNTEREXAMPLE candidate: n={n}, graph_index={idx}, "
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f"source_mode={args.sources}, quantifier={args.quantifier}"
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f"source_mode={args.sources}, quantifier={args.quantifier}, "
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f"restriction={args.restriction}"
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)
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print(f" edges={sorted(tuple(sorted(e)) for e in g.edges(labels=False))}")
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return 1
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@@ -41,19 +41,36 @@
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\newlabel{rem:level-cycle-motivation}{{1.25}{16}}
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\newlabel{def:level-cycle-three-colour-restriction}{{1.26}{16}}
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\newlabel{conj:false-universal-level-cycle-three-colour}{{1.27}{17}}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The $8$-vertex counterexample to the universal-source form. With source $S=\{7\}$, the level cycle $(3,4,5,8)$ lies in $L_2$ and forces all four colours in every proper $4$-vertex-colouring.}}{17}{}\protected@file@percent }
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\newlabel{fig:universal-level-cycle-counterexample}{{6}{17}}
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\newlabel{ex:universal-level-cycle-counterexample}{{1.28}{17}}
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\newlabel{conj:level-cycle-three-colour}{{1.29}{17}}
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\newlabel{def:seam}{{1.30}{17}}
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\newlabel{def:partial-tire-tree}{{1.31}{18}}
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\newlabel{lem:seam-edge-shared}{{1.32}{18}}
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\newlabel{conj:seam-counterexample}{{1.33}{18}}
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\newlabel{conj:level-cycle-three-colour}{{1.29}{18}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{18}{}\protected@file@percent }
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\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Exhaustive vertex-source search for the level-cycle three-colour conjecture on all triangulation isomorphism classes with $4 \leq n \leq 13$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.}}{18}{}\protected@file@percent }
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\newlabel{tab:level-cycle-three-colour-counts}{{1}{18}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{18}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{An inner-boundary refinement}}{18}{}\protected@file@percent }
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\newlabel{def:tire-inner-boundary-three-colour}{{1.30}{18}}
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\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The $5$-connected triangulations at $14 \leq n \leq 24$ generated by \texttt {plantri -c5 -a}. All $9732$ graphs in this slice admit a vertex source witnessing the level-cycle three-colour conjecture.}}{19}{}\protected@file@percent }
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\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{19}}
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\newlabel{conj:tire-inner-boundary-three-colour}{{1.31}{19}}
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\newlabel{rem:inner-boundary-vs-level-cycle}{{1.32}{19}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for the inner-boundary conjecture}}{19}{}\protected@file@percent }
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\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces Exhaustive vertex-source search for the tire inner-boundary three-colour conjecture (Conjecture\nonbreakingspace 1.31\hbox {}) on all triangulation isomorphism classes with $4 \leq n \leq 13$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.}}{20}{}\protected@file@percent }
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\newlabel{tab:inner-boundary-three-colour-counts}{{3}{20}}
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\@writefile{lot}{\contentsline {table}{\numberline {4}{\ignorespaces The $5$-connected triangulations at $14 \leq n \leq 24$ generated by \texttt {plantri -c5 -a}. All $9732$ graphs in this slice admit a vertex source witnessing the tire inner-boundary three-colour conjecture.}}{20}{}\protected@file@percent }
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\newlabel{tab:inner-boundary-three-colour-c5}{{4}{20}}
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\newlabel{def:seam}{{1.33}{20}}
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\newlabel{def:partial-tire-tree}{{1.34}{21}}
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\newlabel{lem:seam-edge-shared}{{1.35}{21}}
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\newlabel{conj:seam-counterexample}{{1.36}{21}}
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\bibcite{tait-original}{1}
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\bibcite{bauerfeld-depth}{2}
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\bibcite{bauerfeld-nested-tire-duals}{3}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent0}{12.7778pt}
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\newlabel{tocindent0}{14.69437pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{19}{}\protected@file@percent }
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\gdef \@abspage@last{19}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{22}{}\protected@file@percent }
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\gdef \@abspage@last{22}
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@@ -1,6 +1,5 @@
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# Fdb version 3
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["pdflatex"] 1780341904 "paper.tex" "paper.pdf" "paper" 1780341906
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||||
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Binary file not shown.
@@ -1348,7 +1348,7 @@ with the level-cycle three-colour restriction with respect to $S$.
|
||||
\subsection*{Enumeration for small $n$}
|
||||
|
||||
We exhaustively enumerated all plane triangulation isomorphism classes with
|
||||
$4 \leq n \leq 12$ vertices and searched the vertex sources for each graph.
|
||||
$4 \leq n \leq 13$ vertices and searched the vertex sources for each graph.
|
||||
No counterexample to Conjecture~\ref{conj:level-cycle-three-colour} appeared
|
||||
in this range. Table~\ref{tab:level-cycle-three-colour-counts} records the
|
||||
size of the search space and the number of triangulations that admit a
|
||||
@@ -1369,11 +1369,187 @@ $9$ & $50$ & $50$ \\
|
||||
$10$ & $233$ & $233$ \\
|
||||
$11$ & $1249$ & $1249$ \\
|
||||
$12$ & $7595$ & $7595$ \\
|
||||
$13$ & $49566$ & $49566$ \\
|
||||
\end{tabular}
|
||||
\caption{Exhaustive vertex-source search for the level-cycle three-colour conjecture on all triangulation isomorphism classes with $4 \leq n \leq 12$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.}
|
||||
\caption{Exhaustive vertex-source search for the level-cycle three-colour conjecture on all triangulation isomorphism classes with $4 \leq n \leq 13$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.}
|
||||
\label{tab:level-cycle-three-colour-counts}
|
||||
\end{table}
|
||||
|
||||
We also tested the six dual triangulations of the Holton--McKay graphs,
|
||||
which lie just beyond this census, and found witnesses in each case.
|
||||
|
||||
\subsection*{The $5$-connected slice at $n \leq 24$}
|
||||
|
||||
As a compact test above the full small-$n$ census, we also enumerated the
|
||||
$5$-connected triangulations at $14 \leq n \leq 24$ with \texttt{plantri
|
||||
-c5 -a}. These are especially rigid triangulations, and the slice remains
|
||||
small enough to check exhaustively. Every graph in this slice admits a
|
||||
vertex source witnessing Conjecture~\ref{conj:level-cycle-three-colour}.
|
||||
|
||||
\begin{table}[ht]
|
||||
\centering
|
||||
\small
|
||||
\setlength{\tabcolsep}{4pt}
|
||||
\begin{tabular}{ccc}
|
||||
$n$ & $5$-connected triangulations & with witness \\\hline
|
||||
$14$ & $1$ & $1$ \\
|
||||
$15$ & $1$ & $1$ \\
|
||||
$16$ & $3$ & $3$ \\
|
||||
$17$ & $4$ & $4$ \\
|
||||
$18$ & $12$ & $12$ \\
|
||||
$19$ & $23$ & $23$ \\
|
||||
$20$ & $71$ & $71$ \\
|
||||
$21$ & $187$ & $187$ \\
|
||||
$22$ & $627$ & $627$ \\
|
||||
$23$ & $1970$ & $1970$ \\
|
||||
$24$ & $6833$ & $6833$ \\
|
||||
\end{tabular}
|
||||
\caption{The $5$-connected triangulations at $14 \leq n \leq 24$ generated by
|
||||
\texttt{plantri -c5 -a}. All $9732$ graphs in this slice admit a vertex
|
||||
source witnessing the level-cycle three-colour conjecture.}
|
||||
\label{tab:level-cycle-three-colour-c5-14-16}
|
||||
\end{table}
|
||||
|
||||
\subsection*{An inner-boundary refinement}
|
||||
|
||||
The level-cycle restriction constrains \emph{every} simple cycle in
|
||||
every level. For the tire-tree program, the cycles that actually carry
|
||||
boundary state are fewer: each tire transfers colour information across
|
||||
its tread between its two boundaries
|
||||
(Theorem~\ref{thm:tire-chromatic-polynomial-transfer}), so it is the tire
|
||||
\emph{inner boundaries} $B_{\mathrm{in}}^{(T)}$ --- not all level cycles
|
||||
--- that one wishes to compress. This motivates a restriction stated
|
||||
directly in the objects of the decomposition.
|
||||
|
||||
\begin{definition}[Tire inner-boundary three-colour restriction]
|
||||
\label{def:tire-inner-boundary-three-colour}
|
||||
Let $G$ be a maximal planar graph, let $v_0 \in V(G)$ be a vertex source
|
||||
on the outer face of $\Pi_G$, and let $c \colon V(G) \to \{1,2,3,4\}$ be
|
||||
a proper $4$-vertex-colouring of $G$. We say $c$ has the \emph{tire
|
||||
inner-boundary three-colour restriction} with respect to
|
||||
$\mathcal{T}(G, \{v_0\})$ if every tire tread $T \in
|
||||
\mathcal{T}(G, \{v_0\})$ satisfies
|
||||
\[
|
||||
|c(V(B_{\mathrm{in}}^{(T)}))| \leq 3,
|
||||
\]
|
||||
i.e.\ the inner boundary of every tire omits at least one of the four
|
||||
colours. (A degenerate inner boundary is a single vertex and the
|
||||
condition is then vacuous.)
|
||||
\end{definition}
|
||||
|
||||
\begin{conjecture}[Tire inner-boundary three-colour conjecture]
|
||||
\label{conj:tire-inner-boundary-three-colour}
|
||||
Every maximal planar graph $G$ admits a vertex source $v_0 \in V(G)$ and
|
||||
a proper $4$-vertex-colouring $c$ of $G$ such that $c$ has the tire
|
||||
inner-boundary three-colour restriction with respect to
|
||||
$\mathcal{T}(G, \{v_0\})$.
|
||||
\end{conjecture}
|
||||
|
||||
\begin{remark}[Relation to the level-cycle conjecture]
|
||||
\label{rem:inner-boundary-vs-level-cycle}
|
||||
For a depth-$d$ tire $T$, the inner outerplanar graph satisfies
|
||||
$O^{(T)} \subseteq G[L_{d+1}]$: a depth-$d$ face has its three vertex
|
||||
levels in $\{d, d+1\}$ (adjacent vertices differ by at most one level),
|
||||
so the level-$(d+1)$ vertices of the tire's dual component are exactly
|
||||
$V(O^{(T)})$. Since $O^{(T)}$ is outerplanar, every one of its vertices
|
||||
lies on the inner-boundary walk, whence $V(B_{\mathrm{in}}^{(T)}) =
|
||||
V(O^{(T)}) \subseteq L_{d+1}$ is supported on a single level, and is a
|
||||
simple level cycle when $O^{(T)}$ is $2$-connected.
|
||||
|
||||
Consequently the vertex-source form of
|
||||
Conjecture~\ref{conj:level-cycle-three-colour} implies
|
||||
Conjecture~\ref{conj:tire-inner-boundary-three-colour} on every
|
||||
$2$-connected inner boundary: the witnessing colouring already makes
|
||||
each such cycle omit a colour. The present conjecture is thus a
|
||||
\emph{weakening}, constraining only the inner-boundary cycles of one
|
||||
tire-tree decomposition rather than all level cycles of some level
|
||||
source. It is no harder than the vertex-source form of
|
||||
Conjecture~\ref{conj:level-cycle-three-colour}, while targeting exactly
|
||||
the interface the chromatic-transfer machinery of
|
||||
Theorem~\ref{thm:tire-chromatic-polynomial-transfer} runs across.
|
||||
(The non-$2$-connected case --- an
|
||||
inner boundary whose walk traverses a bridge or cut-vertex of $O^{(T)}$
|
||||
--- is not covered by the simple-cycle statement of
|
||||
Conjecture~\ref{conj:level-cycle-three-colour} and must be argued
|
||||
separately.)
|
||||
\end{remark}
|
||||
|
||||
\subsection*{Enumeration for the inner-boundary conjecture}
|
||||
|
||||
We repeated the exhaustive search of
|
||||
Conjecture~\ref{conj:level-cycle-three-colour} for the inner-boundary
|
||||
restriction, testing for each triangulation whether some vertex source
|
||||
$v_0$ admits a proper $4$-colouring whose tire inner boundaries each omit
|
||||
a colour. For a depth-$d$ tire the inner-boundary vertex set is computed
|
||||
directly as the level-$(d+1)$ vertices of the corresponding depth-$d$
|
||||
dual component, using
|
||||
Remark~\ref{rem:inner-boundary-vs-level-cycle}. No counterexample
|
||||
appeared on the full small-$n$ census $4 \leq n \leq 13$
|
||||
(Table~\ref{tab:inner-boundary-three-colour-counts}) or on the
|
||||
$5$-connected slice $14 \leq n \leq 24$
|
||||
(Table~\ref{tab:inner-boundary-three-colour-c5}).
|
||||
|
||||
\begin{table}[ht]
|
||||
\centering
|
||||
\small
|
||||
\setlength{\tabcolsep}{4pt}
|
||||
\begin{tabular}{ccc}
|
||||
$n$ & triangulations & with witness \\\hline
|
||||
$4$ & $1$ & $1$ \\
|
||||
$5$ & $1$ & $1$ \\
|
||||
$6$ & $2$ & $2$ \\
|
||||
$7$ & $5$ & $5$ \\
|
||||
$8$ & $14$ & $14$ \\
|
||||
$9$ & $50$ & $50$ \\
|
||||
$10$ & $233$ & $233$ \\
|
||||
$11$ & $1249$ & $1249$ \\
|
||||
$12$ & $7595$ & $7595$ \\
|
||||
$13$ & $49566$ & $49566$ \\
|
||||
\end{tabular}
|
||||
\caption{Exhaustive vertex-source search for the tire inner-boundary
|
||||
three-colour conjecture
|
||||
(Conjecture~\ref{conj:tire-inner-boundary-three-colour}) on all
|
||||
triangulation isomorphism classes with $4 \leq n \leq 13$. Every
|
||||
triangulation in this range admits at least one vertex source
|
||||
witnessing the conjecture.}
|
||||
\label{tab:inner-boundary-three-colour-counts}
|
||||
\end{table}
|
||||
|
||||
\begin{table}[ht]
|
||||
\centering
|
||||
\small
|
||||
\setlength{\tabcolsep}{4pt}
|
||||
\begin{tabular}{ccc}
|
||||
$n$ & $5$-connected triangulations & with witness \\\hline
|
||||
$14$ & $1$ & $1$ \\
|
||||
$15$ & $1$ & $1$ \\
|
||||
$16$ & $3$ & $3$ \\
|
||||
$17$ & $4$ & $4$ \\
|
||||
$18$ & $12$ & $12$ \\
|
||||
$19$ & $23$ & $23$ \\
|
||||
$20$ & $71$ & $71$ \\
|
||||
$21$ & $187$ & $187$ \\
|
||||
$22$ & $627$ & $627$ \\
|
||||
$23$ & $1970$ & $1970$ \\
|
||||
$24$ & $6833$ & $6833$ \\
|
||||
\end{tabular}
|
||||
\caption{The $5$-connected triangulations at $14 \leq n \leq 24$
|
||||
generated by \texttt{plantri -c5 -a}. All $9732$ graphs in this slice
|
||||
admit a vertex source witnessing the tire inner-boundary three-colour
|
||||
conjecture.}
|
||||
\label{tab:inner-boundary-three-colour-c5}
|
||||
\end{table}
|
||||
|
||||
Unlike the small-$n$ census, where the first source and colouring tried
|
||||
typically already witness the restriction, the source choice is
|
||||
genuinely active in the $5$-connected slice: many vertex sources fail
|
||||
exhaustively before a witness is found. For instance, in the unique
|
||||
$n=16$ $5$-connected triangulation two sources exhaust all proper
|
||||
$4$-colourings with no compatible colouring before a third source
|
||||
succeeds. This is consistent with the failure of the universal-source
|
||||
form (Conjecture~\ref{conj:false-universal-level-cycle-three-colour}):
|
||||
the existential quantifier over the root is doing real work.
|
||||
|
||||
\begin{definition}[Seam]
|
||||
\label{def:seam}
|
||||
A \emph{seam} of a maximal planar graph $G$ is a simple cycle
|
||||
|
||||
Reference in New Issue
Block a user