diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_inner_boundary_counterexample.py b/papers/coloring_nested_tire_graphs/experiments/draw_inner_boundary_counterexample.py new file mode 100644 index 0000000..24ca2f3 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/draw_inner_boundary_counterexample.py @@ -0,0 +1,110 @@ +"""Draw the 14-vertex counterexample to the tire inner-boundary +three-colour conjecture (Conjecture~\\ref{conj:tire-inner-boundary-three-colour}). + +This is the graph at index 263993 in the n=14 plantri enumeration: a +3-connected (but not 5-connected) maximal planar graph with degree +sequence [7,7,7,7,7,7,6,6,3,3,3,3,3,3] and exactly 96 proper +4-colourings, none of which witness the inner-boundary restriction +from any vertex source. +""" + +from __future__ import annotations + +import os + +import matplotlib.pyplot as plt +import networkx as nx +from matplotlib.lines import Line2D + + +OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__))) + + +EDGES = [ + (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), + (2, 3), (2, 4), (2, 6), (2, 8), (2, 9), (2, 10), + (3, 4), (4, 5), (4, 6), (4, 10), + (5, 6), (6, 7), (6, 9), (6, 10), + (7, 8), (7, 9), (7, 11), (7, 12), (7, 13), + (8, 9), (8, 12), (8, 13), (8, 14), + (9, 11), (9, 12), (9, 14), + (11, 12), (12, 13), (12, 14), +] + + +def build_graph() -> nx.Graph: + g = nx.Graph() + g.add_edges_from(EDGES) + return g + + +def main() -> int: + g = build_graph() + is_planar, _ = nx.check_planarity(g) + if not is_planar: + raise RuntimeError("graph should be planar") + pos = nx.planar_layout(g, scale=3.4) + + fig, ax = plt.subplots(figsize=(8.2, 7.4)) + + nx.draw_networkx_edges(g, pos, ax=ax, edge_color="#cbd5e1", width=1.3) + + deg3 = [v for v in g.nodes() if g.degree(v) == 3] + deg7 = [v for v in g.nodes() if g.degree(v) == 7] + other = [v for v in g.nodes() if v not in deg3 and v not in deg7] + + def draw_nodes(nodes, color): + for v in nodes: + x, y = pos[v] + ax.scatter( + [x], [y], + s=520, + color=color, + edgecolors="black", + linewidths=1.1, + zorder=3, + ) + ax.text( + x, y, f"{v}", + ha="center", va="center", + color="white", fontsize=11, fontweight="bold", + zorder=4, + ) + + draw_nodes(deg7, "#0f172a") + draw_nodes(other, "#475569") + draw_nodes(deg3, "#94a3b8") + + legend = [ + Line2D([0], [0], marker="o", color="w", + label=r"degree $7$", + markerfacecolor="#0f172a", markeredgecolor="black", + markersize=12), + Line2D([0], [0], marker="o", color="w", + label=r"degree $6$", + markerfacecolor="#475569", markeredgecolor="black", + markersize=12), + Line2D([0], [0], marker="o", color="w", + label=r"degree $3$", + markerfacecolor="#94a3b8", markeredgecolor="black", + markersize=12), + ] + ax.legend(handles=legend, loc="lower center", ncol=3, framealpha=0.95) + ax.set_title( + "Counterexample to the inner-boundary three-colour conjecture " + "($n=14$)", + fontsize=13, pad=14, + ) + ax.set_aspect("equal") + ax.axis("off") + fig.tight_layout(rect=[0, 0.05, 1, 1]) + + out = os.path.join(OUT_DIR, "fig_inner_boundary_counterexample.png") + fig.savefig(out, dpi=180, bbox_inches="tight") + plt.close(fig) + print(f"wrote {out}") + return 0 + + +if __name__ == "__main__": + raise SystemExit(main()) diff --git a/papers/coloring_nested_tire_graphs/fig_inner_boundary_counterexample.png b/papers/coloring_nested_tire_graphs/fig_inner_boundary_counterexample.png new file mode 100644 index 0000000..11da609 Binary files /dev/null and b/papers/coloring_nested_tire_graphs/fig_inner_boundary_counterexample.png differ diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index c348274..9c6133f 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -44,26 +44,26 @@ \@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}{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}} +\newlabel{def:tire-inner-boundary-three-colour}{{1.29}{18}} +\newlabel{conj:tire-inner-boundary-three-colour}{{1.30}{18}} +\newlabel{rem:inner-boundary-vs-level-cycle}{{1.31}{18}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A counterexample at $n=14$}}{18}{}\protected@file@percent } +\newlabel{ex:inner-boundary-counterexample}{{1.32}{18}} +\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces The $14$-vertex counterexample $G^\star $ to Conjecture\nonbreakingspace 1.30\hbox {} in a planar embedding. The six degree-$3$ vertices split into two triples, $\{3,5,10\}$ each adjacent to a triangle in the core $\{1,2,4,6\}$, and $\{11,13,14\}$ each adjacent to a triangle in the core $\{7,8,9,12\}$; the two cores are joined by the edges $17,28,69$ together with $12$.}}{19}{}\protected@file@percent } +\newlabel{fig:inner-boundary-counterexample}{{7}{19}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{20}{}\protected@file@percent } +\newlabel{conj:level-cycle-three-colour}{{1.33}{20}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{20}{}\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.}}{20}{}\protected@file@percent } +\newlabel{tab:level-cycle-three-colour-counts}{{1}{20}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{20}{}\protected@file@percent } +\newlabel{def:seam}{{1.34}{20}} +\@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.}}{21}{}\protected@file@percent } +\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{21}} +\newlabel{def:partial-tire-tree}{{1.35}{21}} +\newlabel{lem:seam-edge-shared}{{1.36}{21}} +\newlabel{conj:seam-counterexample}{{1.37}{21}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} diff --git a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk index 5d469ed..7bf8e49 100644 --- a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk +++ b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1780361150 "paper.tex" "paper.pdf" "paper" 1780361151 +["pdflatex"] 1780365355 "paper.tex" "paper.pdf" "paper" 1780365356 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -143,11 +143,12 @@ "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" "fig_dual_depth.png" 1779857443 255786 cb48aab5aa40fc161d13a75df0544511 "" + "fig_inner_boundary_counterexample.png" 1780363332 86866 e15e4311e42fec20179ac6bb90683dea "" "fig_tire_example.png" 1779857443 104494 8f9ce26b469b4236b8b67829f73a5faa "" "fig_tire_tree_decomposition.png" 1780290287 372371 1b44f5a3e9f637d78ae951b1f2e3a89d "" "fig_universal_level_cycle_counterexample.png" 1780325973 75145 08f600be4e05c11d702bee45996ca222 "" - "paper.aux" 1780361151 8992 aa580f9d36e55b0f1c12ef76f2e58090 "pdflatex" - "paper.tex" 1780361128 81166 412ad5cc268416100823025ff610b3f4 "" + "paper.aux" 1780365356 8877 506e0c8fb0cf3332db2191566c33d175 "pdflatex" + "paper.tex" 1780365326 81811 cdae7a96d08d9776f308c7c25eefa561 "" (generated) "paper.aux" "paper.log" diff --git a/papers/coloring_nested_tire_graphs/paper.fls b/papers/coloring_nested_tire_graphs/paper.fls index d72e699..0882749 100644 --- a/papers/coloring_nested_tire_graphs/paper.fls +++ b/papers/coloring_nested_tire_graphs/paper.fls @@ -478,6 +478,10 @@ INPUT fig_universal_level_cycle_counterexample.png INPUT ./fig_universal_level_cycle_counterexample.png INPUT ./fig_universal_level_cycle_counterexample.png INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm +INPUT ./fig_inner_boundary_counterexample.png +INPUT fig_inner_boundary_counterexample.png +INPUT ./fig_inner_boundary_counterexample.png +INPUT ./fig_inner_boundary_counterexample.png INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm INPUT paper.aux INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index b19dbfa..c5d55f5 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 1 JUN 2026 20:45 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 1 JUN 2026 21:55 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -541,53 +541,64 @@ File: fig_universal_level_cycle_counterexample.png Graphic file (type png) Package pdftex.def Info: fig_universal_level_cycle_counterexample.png used on input line 1303. 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Therefore no proper $4$-colouring has the level-cycle three-colour restriction with respect to $S=\{7\}$. \end{example} +\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*{A counterexample at $n=14$} + +Conjecture~\ref{conj:tire-inner-boundary-three-colour} is in fact +false. An exhaustive search over the triangulations enumerated by +\texttt{plantri} at $n=14$ encounters a graph $G^\star$ on $14$ vertices +and $36$ edges --- specifically, the graph at index $263993$ in the +\texttt{plantri} enumeration --- for which no vertex source admits any +witness. + +\begin{example}[Counterexample to Conjecture~\ref{conj:tire-inner-boundary-three-colour}] +\label{ex:inner-boundary-counterexample} +Let $G^\star$ be the maximal planar graph with vertex set +$\{1,2,\dots,14\}$ and edge set +\begin{align*} +E(G^\star) = \{ + & 12, 13, 14, 15, 16, 17, 18, \\ + & 23, 24, 26, 28, 29, 2\,10, \\ + & 34, 45, 46, 4\,10, 56, 67, 69, 6\,10, \\ + & 78, 79, 7\,11, 7\,12, 7\,13, \\ + & 89, 8\,12, 8\,13, 8\,14, \\ + & 9\,11, 9\,12, 9\,14, \\ + & 11\,12, 12\,13, 12\,14 +\}. +\end{align*} +The graph $G^\star$ is a $3$-connected (but not $5$-connected) planar +triangulation with degree sequence +$(7,7,7,7,7,7,6,6,3,3,3,3,3,3)$ and exactly $96$ proper $4$-vertex +colourings. For \emph{every} choice of vertex source +$v_0 \in V(G^\star)$, each of the $96$ proper $4$-colourings of +$G^\star$ has some tire whose inner boundary uses all four colours. +A planar embedding is shown in +Figure~\ref{fig:inner-boundary-counterexample}. +\end{example} + +\begin{figure}[ht] +\centering +\includegraphics[width=0.78\textwidth]{fig_inner_boundary_counterexample} +\caption{The $14$-vertex counterexample $G^\star$ to +Conjecture~\ref{conj:tire-inner-boundary-three-colour} in a planar +embedding. The six degree-$3$ vertices split into two triples, +$\{3,5,10\}$ each adjacent to a triangle in the +core $\{1,2,4,6\}$, and $\{11,13,14\}$ each adjacent to a triangle in +the core $\{7,8,9,12\}$; the two cores are joined by the edges +$17,28,69$ together with $12$.} +\label{fig:inner-boundary-counterexample} +\end{figure} + +The failure was verified by enumerating, for each of the $14$ vertex +sources, all $96$ proper $4$-colourings of $G^\star$ and computing the +inner boundary $V(B_{\mathrm{in}}^{(T)})$ of every tire $T$ as the +level-$(d+1)$ vertices of the corresponding depth-$d$ dual component +(Remark~\ref{rem:inner-boundary-vs-level-cycle}). Each source has +exactly two non-degenerate inner boundaries (size $\geq 4$), and every +proper $4$-colouring assigns all four colours to at least one of them. + +Because Conjecture~\ref{conj:tire-inner-boundary-three-colour} is a +weakening of the vertex-source form of +Conjecture~\ref{conj:level-cycle-three-colour} only \emph{on +$2$-connected inner boundaries} +(Remark~\ref{rem:inner-boundary-vs-level-cycle}), $G^\star$ need not +refute Conjecture~\ref{conj:level-cycle-three-colour}, and in fact does +not: the vertex source $v_0 = 10$ admits a proper $4$-colouring under +which every simple level cycle uses at most three colours. Combining +this with Remark~\ref{rem:inner-boundary-vs-level-cycle}, at least one +tire $T$ under $v_0 = 10$ must have an inner outerplanar graph +$O^{(T)}$ that fails to be $2$-connected --- under the witnessing +colouring, some inner boundary uses all four colours, and if its +$O^{(T)}$ were $2$-connected then by +Remark~\ref{rem:inner-boundary-vs-level-cycle} that inner boundary +would be a simple level cycle, contradicting the level-cycle witness. +The failure of +Conjecture~\ref{conj:tire-inner-boundary-three-colour} is therefore +attributable to the non-$2$-connected case left open by +Remark~\ref{rem:inner-boundary-vs-level-cycle}, not to a deeper failure +of the level-cycle statement on $G^\star$. + +\subsection*{The surviving level-cycle conjecture} + +The verification on $G^\star$ above is consistent with the broader +empirical picture for the level-cycle restriction, which we record as +the conjecture this section ultimately stands on. + \begin{conjecture}[Level-cycle three-colour conjecture] \label{conj:level-cycle-three-colour} Let $G$ be a maximal planar graph. Then there exists a level source @@ -1410,153 +1556,6 @@ 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} - -We also re-ran the inner-boundary check on the six dual triangulations -of the Holton--McKay graphs and found a vertex source witnessing the -conjecture for each. In four of the six cases the first source tried -already succeeded; in the remaining two, one or two vertex sources -exhausted all $4320$ proper $4$-colourings before a witnessing source -was found. - -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