From 88c74efd28919d43928f40596e7b2a58f017eaf6 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 1 Jun 2026 15:29:34 -0400 Subject: [PATCH] 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 --- .../check_level_cycle_three_color.py | 185 +++++++++- papers/coloring_nested_tire_graphs/paper.aux | 33 +- .../paper.fdb_latexmk | 10 +- papers/coloring_nested_tire_graphs/paper.fls | 9 +- papers/coloring_nested_tire_graphs/paper.log | 341 +++++++++++++----- papers/coloring_nested_tire_graphs/paper.pdf | Bin 1003974 -> 1022633 bytes papers/coloring_nested_tire_graphs/paper.tex | 180 ++++++++- 7 files changed, 637 insertions(+), 121 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/experiments/check_level_cycle_three_color.py b/papers/coloring_nested_tire_graphs/experiments/check_level_cycle_three_color.py index d421c1b..bc7a9e4 100644 --- a/papers/coloring_nested_tire_graphs/experiments/check_level_cycle_three_color.py +++ b/papers/coloring_nested_tire_graphs/experiments/check_level_cycle_three_color.py @@ -1,24 +1,34 @@ -"""Empirical test for the level-cycle three-colour conjecture. +"""Empirical test for the level-cycle three-colour conjecture and its +tire inner-boundary refinement. -The weakened conjecture says: for every maximal planar graph G, there -is some level source S and some proper 4-vertex-colouring c such that -every simple cycle contained in a single level G[L_d] uses at most -three colours. +The level-cycle conjecture (``--restriction level-cycle``, the default) +says: for every maximal planar graph G, there is some level source S and +some proper 4-vertex-colouring c such that every simple cycle contained +in a single level G[L_d] uses at most three colours. -Important: this checks the cycle-by-cycle version. Two cycles in the -same level or the same inner outerplanar component may omit different -colours. +The inner-boundary conjecture (``--restriction inner-boundary``) is the +weakening that constrains only the inner boundaries B_in of the tires in +a tire-tree decomposition rooted at a vertex source: for every G there is +a vertex source v0 and a proper 4-colouring c such that every tire inner +boundary uses at most three colours. Because the inner outerplanar graph +O of a depth-d tire satisfies O = G[component faces] cap L_{d+1} and is +outerplanar, V(B_in) is exactly the level-(d+1) vertices of the tire's +dual component (see the inner-boundary remark in the paper). + +Important: both checks are cycle-by-cycle. Two cycles in the same level +or the same inner outerplanar component may omit different colours. Run examples: sage -python experiments/check_level_cycle_three_color.py 4 9 sage -python experiments/check_level_cycle_three_color.py 4 8 --quantifier all-sources sage -python experiments/check_level_cycle_three_color.py 4 8 --sources all + sage -python experiments/check_level_cycle_three_color.py 4 9 --restriction inner-boundary """ from __future__ import annotations import argparse -from collections import deque +from collections import defaultdict, deque from itertools import combinations from typing import Any, Iterable, Iterator, Sequence, cast @@ -131,10 +141,93 @@ def level_cycle_violation( return None +def inner_boundary_vertex_sets(g: Graph, source: Source) -> list[frozenset[Any]]: + """Inner-boundary vertex sets of the tire-tree decomposition at a vertex. + + The tire-tree decomposition is defined only for a single-vertex source + placed on the outer face. Each tire is a connected component of the + depth-d dual subgraph G'_d (faces whose minimum vertex level is d). Because + a depth-d face has its three vertex levels in {d, d+1} (adjacent vertices + differ by at most one level), and because the inner outerplanar graph O of + the tire is outerplanar (so every vertex of O lies on its inner-boundary + walk), the inner-boundary vertex set equals the level-(d+1) vertices of the + component: V(B_in) = V(O) = V(component) cap L_{d+1}. + + We work on the sphere dual (all faces). The choice of outer face does not + change any inner-boundary set: depth->=1 components are untouched by removing + a depth-0 face, and the root tire always has O = G[L_1]. + + Only sets of size >= 4 are returned; smaller sets can never use four colours. + """ + if len(source) != 1: + raise ValueError("inner-boundary restriction requires a vertex source") + distances = distances_from_source(g, source) + + emb = cast(Graph, g.copy()) + if not emb.is_planar(set_embedding=True): + raise ValueError("graph is not planar; cannot build tire decomposition") + faces = emb.faces() + + face_vertices: list[set[Any]] = [] + for face in faces: + verts: set[Any] = set() + for edge in face: + verts.add(edge[0]) + verts.add(edge[1]) + face_vertices.append(verts) + depths = [min(distances[v] for v in verts) for verts in face_vertices] + + edge_faces: dict[frozenset[Any], list[int]] = defaultdict(list) + for i, face in enumerate(faces): + for edge in face: + edge_faces[frozenset((edge[0], edge[1]))].append(i) + dual_adj: dict[int, set[int]] = defaultdict(set) + for incident in edge_faces.values(): + for a in range(len(incident)): + for b in range(a + 1, len(incident)): + dual_adj[incident[a]].add(incident[b]) + dual_adj[incident[b]].add(incident[a]) + + targets: list[frozenset[Any]] = [] + seen = [False] * len(faces) + for start in range(len(faces)): + if seen[start]: + continue + depth = depths[start] + component = [start] + seen[start] = True + stack = [start] + while stack: + f = stack.pop() + for h in dual_adj[f]: + if not seen[h] and depths[h] == depth: + seen[h] = True + component.append(h) + stack.append(h) + inner: set[Any] = set() + for f in component: + inner.update(v for v in face_vertices[f] if distances[v] == depth + 1) + if len(inner) >= 4: + targets.append(frozenset(inner)) + return targets + + +def first_inner_boundary_violation( + targets: Sequence[frozenset[Any]], coloring: Coloring +) -> tuple[frozenset[Any], set[int]] | None: + """Return the first inner boundary using all four colours, if any.""" + for inner in targets: + used = {coloring[v] for v in inner} + if len(used) > 3: + return inner, used + return None + + def coloring_witness( g: Graph, source: Source, max_colorings: int | None, + restriction: str, ) -> tuple[Coloring | None, int, bool]: """Find a proper 4-colouring satisfying the conjectured restriction. @@ -142,11 +235,20 @@ def coloring_witness( exhausted is False, max_colorings was reached before a decision. """ distances = distances_from_source(g, source) + targets = ( + inner_boundary_vertex_sets(g, source) + if restriction == "inner-boundary" + else None + ) checked = 0 for raw in all_graph_colorings(g, 4, vertex_color_dict=True): coloring = cast(Coloring, raw) checked += 1 - if level_cycle_violation(g, distances, coloring) is None: + if restriction == "inner-boundary": + violated = first_inner_boundary_violation(cast(Sequence, targets), coloring) + else: + violated = level_cycle_violation(g, distances, coloring) + if violated is None: return coloring, checked, True if max_colorings is not None and checked >= max_colorings: return None, checked, False @@ -165,6 +267,7 @@ def test_graph( max_colorings: int | None, stop_first: bool, quantifier: str, + restriction: str, ) -> tuple[bool, bool, int]: """Test a graph over selected sources. @@ -175,7 +278,9 @@ def test_graph( found_any_source = False for source in sources: checked_sources += 1 - witness, n_checked, exhausted = coloring_witness(g, source, max_colorings) + witness, n_checked, exhausted = coloring_witness( + g, source, max_colorings, restriction + ) if witness is None: if not exhausted: complete = False @@ -185,11 +290,16 @@ def test_graph( f"colorings_checked={n_checked}" ) if exhausted and quantifier == "all-sources": - distances = distances_from_source(g, source) - first = next(all_graph_colorings(g, 4, vertex_color_dict=True), None) - if first is not None: - violation = level_cycle_violation(g, distances, cast(Coloring, first)) - print(f" first_coloring_violation={violation}") + if restriction == "level-cycle": + distances = distances_from_source(g, source) + first = next( + all_graph_colorings(g, 4, vertex_color_dict=True), None + ) + if first is not None: + violation = level_cycle_violation( + g, distances, cast(Coloring, first) + ) + print(f" first_coloring_violation={violation}") return False, complete, checked_sources if stop_first: if quantifier == "all-sources": @@ -232,12 +342,31 @@ def parse_args() -> argparse.Namespace: "the stronger earlier version" ), ) + parser.add_argument( + "--restriction", + choices=("level-cycle", "inner-boundary"), + default="level-cycle", + help=( + "level-cycle constrains every simple level cycle; inner-boundary " + "constrains only the tire inner boundaries of a vertex-rooted " + "tire-tree decomposition" + ), + ) parser.add_argument( "--max-cycle-source-size", type=int, default=None, help="optional cap on induced cycle source size", ) + parser.add_argument( + "--min-connectivity", + type=int, + default=None, + help=( + "restrict to triangulations of at least this vertex connectivity " + "(e.g. 5 for the 5-connected slice); passed to plantri via Sage" + ), + ) parser.add_argument( "--max-colorings", type=int, @@ -255,20 +384,37 @@ def parse_args() -> argparse.Namespace: def main() -> int: args = parse_args() stop_first = not args.full + if args.restriction == "inner-boundary" and args.sources != "vertex": + print( + "note: inner-boundary restriction is vertex-rooted; " + f"overriding --sources {args.sources} with vertex" + ) + args.sources = "vertex" total_graphs = 0 total_sources = 0 unknown = 0 + triangulation_kwargs = {} + if args.min_connectivity is not None: + triangulation_kwargs["minimum_connectivity"] = args.min_connectivity + for n in range(args.n_min, args.n_max + 1): print(f"=== n={n} ===") - for idx, g in enumerate(graphs.triangulations(n), start=1): + for idx, g in enumerate( + graphs.triangulations(n, **triangulation_kwargs), start=1 + ): total_graphs += 1 source_list = list( level_sources(g, args.sources, args.max_cycle_source_size) ) print(f" graph #{idx}: sources={len(source_list)}") passed, complete, checked_sources = test_graph( - g, source_list, args.max_colorings, stop_first, args.quantifier + g, + source_list, + args.max_colorings, + stop_first, + args.quantifier, + args.restriction, ) total_sources += checked_sources if not complete: @@ -276,7 +422,8 @@ def main() -> int: if not passed: print( f"COUNTEREXAMPLE candidate: n={n}, graph_index={idx}, " - f"source_mode={args.sources}, quantifier={args.quantifier}" + f"source_mode={args.sources}, quantifier={args.quantifier}, " + f"restriction={args.restriction}" ) print(f" edges={sorted(tuple(sorted(e)) for e in g.edges(labels=False))}") return 1 diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 11196e0..c348274 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -41,19 +41,36 @@ \newlabel{rem:level-cycle-motivation}{{1.25}{16}} \newlabel{def:level-cycle-three-colour-restriction}{{1.26}{16}} \newlabel{conj:false-universal-level-cycle-three-colour}{{1.27}{17}} +\@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 } +\newlabel{fig:universal-level-cycle-counterexample}{{6}{17}} \newlabel{ex:universal-level-cycle-counterexample}{{1.28}{17}} -\newlabel{conj:level-cycle-three-colour}{{1.29}{17}} -\newlabel{def:seam}{{1.30}{17}} -\newlabel{def:partial-tire-tree}{{1.31}{18}} -\newlabel{lem:seam-edge-shared}{{1.32}{18}} -\newlabel{conj:seam-counterexample}{{1.33}{18}} +\newlabel{conj:level-cycle-three-colour}{{1.29}{18}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{18}{}\protected@file@percent } +\@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 } +\newlabel{tab:level-cycle-three-colour-counts}{{1}{18}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{18}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{An inner-boundary refinement}}{18}{}\protected@file@percent } +\newlabel{def:tire-inner-boundary-three-colour}{{1.30}{18}} +\@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 } +\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{19}} +\newlabel{conj:tire-inner-boundary-three-colour}{{1.31}{19}} +\newlabel{rem:inner-boundary-vs-level-cycle}{{1.32}{19}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for the inner-boundary conjecture}}{19}{}\protected@file@percent } +\@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 } +\newlabel{tab:inner-boundary-three-colour-counts}{{3}{20}} +\@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 } +\newlabel{tab:inner-boundary-three-colour-c5}{{4}{20}} +\newlabel{def:seam}{{1.33}{20}} +\newlabel{def:partial-tire-tree}{{1.34}{21}} +\newlabel{lem:seam-edge-shared}{{1.35}{21}} +\newlabel{conj:seam-counterexample}{{1.36}{21}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} \newlabel{tocindent-1}{0pt} -\newlabel{tocindent0}{12.7778pt} +\newlabel{tocindent0}{14.69437pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{19}{}\protected@file@percent } -\gdef \@abspage@last{19} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{22}{}\protected@file@percent } +\gdef \@abspage@last{22} diff --git a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk index 3c289cb..aa0767a 100644 --- a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk +++ b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk @@ -1,6 +1,5 @@ # Fdb version 3 -["pdflatex"] 1780293652 "/Users/didericis/Code/math-research/papers/coloring_nested_tire_graphs/paper.tex" "paper.pdf" "paper" 1780293654 - 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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