From 3910bc9fb7bbcd48e13779dc76ae07e7d693c95f Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 8 Jun 2026 14:56:13 -0400 Subject: [PATCH] Split three-colour restrictions into separate paper --- papers/coloring_nested_tire_graphs/paper.aux | 39 +- .../paper.fdb_latexmk | 11 +- papers/coloring_nested_tire_graphs/paper.fls | 12 - papers/coloring_nested_tire_graphs/paper.log | 113 ++-- papers/coloring_nested_tire_graphs/paper.pdf | Bin 1252642 -> 1096120 bytes papers/coloring_nested_tire_graphs/paper.tex | 408 +------------ .../fig_inner_boundary_counterexample.png | Bin 0 -> 86866 bytes ...g_universal_level_cycle_counterexample.png | Bin 0 -> 75145 bytes .../paper.aux | 64 ++ .../paper.fdb_latexmk | 145 +++++ .../paper.fls | 489 +++++++++++++++ .../paper.log | 555 +++++++++++++++++ .../paper.pdf | Bin 0 -> 387544 bytes .../paper.tex | 566 ++++++++++++++++++ 14 files changed, 1886 insertions(+), 516 deletions(-) create mode 100644 papers/three_color_restrictions_nested_tire_graphs/fig_inner_boundary_counterexample.png create mode 100644 papers/three_color_restrictions_nested_tire_graphs/fig_universal_level_cycle_counterexample.png create mode 100644 papers/three_color_restrictions_nested_tire_graphs/paper.aux create mode 100644 papers/three_color_restrictions_nested_tire_graphs/paper.fdb_latexmk create mode 100644 papers/three_color_restrictions_nested_tire_graphs/paper.fls create mode 100644 papers/three_color_restrictions_nested_tire_graphs/paper.log create mode 100644 papers/three_color_restrictions_nested_tire_graphs/paper.pdf create mode 100644 papers/three_color_restrictions_nested_tire_graphs/paper.tex diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 1d23887..f8960e4 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -8,15 +8,15 @@ \citation{dvorak-lidicky-cones} \citation{heesch-untersuchungen} \citation{robertson-sanders-seymour-thomas} -\citation{robertson-sanders-seymour-thomas} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent } +\citation{robertson-sanders-seymour-thomas} \newlabel{def:dual}{{1.3}{2}} \newlabel{def:dual-depth}{{1.4}{2}} \newlabel{def:dual-component}{{1.5}{2}} -\newlabel{def:tire-graph}{{1.6}{2}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{3}{}\protected@file@percent } \newlabel{fig:dual-depth}{{1}{3}} +\newlabel{def:tire-graph}{{1.6}{3}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{4}{}\protected@file@percent } \newlabel{fig:tire-example}{{2}{4}} \newlabel{def:medial-tire-graph}{{1.7}{4}} @@ -49,34 +49,6 @@ \newlabel{thm:tread-tree}{{1.23}{14}} \newlabel{rem:tree-multiple-children}{{1.24}{15}} \newlabel{thm:tire-tree-decomposition}{{1.25}{15}} -\newlabel{rem:tree-coloring-factorisation}{{1.26}{17}} -\newlabel{rem:level-cycle-motivation}{{1.27}{17}} -\newlabel{def:level-cycle-three-colour-restriction}{{1.28}{17}} -\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.25\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{18}{}\protected@file@percent } -\newlabel{fig:tire-tree-decomposition}{{6}{18}} -\newlabel{conj:false-universal-level-cycle-three-colour}{{1.29}{18}} -\newlabel{ex:universal-level-cycle-counterexample}{{1.30}{18}} -\@writefile{lof}{\contentsline {figure}{\numberline {7}{\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.}}{19}{}\protected@file@percent } -\newlabel{fig:universal-level-cycle-counterexample}{{7}{19}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{An inner-boundary refinement}}{19}{}\protected@file@percent } -\newlabel{def:tire-inner-boundary-three-colour}{{1.31}{19}} -\newlabel{conj:tire-inner-boundary-three-colour}{{1.32}{19}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A counterexample at $n=14$}}{20}{}\protected@file@percent } -\newlabel{ex:inner-boundary-counterexample}{{1.33}{20}} -\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces The $14$-vertex counterexample $G^\star $ to Conjecture\nonbreakingspace 1.32\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 $12$.}}{20}{}\protected@file@percent } -\newlabel{fig:inner-boundary-counterexample}{{8}{20}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{21}{}\protected@file@percent } -\newlabel{conj:level-cycle-three-colour}{{1.34}{21}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{21}{}\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.}}{21}{}\protected@file@percent } -\newlabel{tab:level-cycle-three-colour-counts}{{1}{21}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{21}{}\protected@file@percent } -\newlabel{def:seam}{{1.35}{21}} -\@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.}}{22}{}\protected@file@percent } -\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{22}} -\newlabel{def:partial-tire-tree}{{1.36}{22}} -\newlabel{lem:seam-edge-shared}{{1.37}{22}} -\newlabel{conj:seam-counterexample}{{1.38}{22}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} @@ -93,5 +65,8 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{23}{}\protected@file@percent } -\gdef \@abspage@last{23} +\newlabel{rem:tree-coloring-factorisation}{{1.26}{17}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{17}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.25\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{18}{}\protected@file@percent } +\newlabel{fig:tire-tree-decomposition}{{6}{18}} +\gdef \@abspage@last{18} diff --git a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk index cba488c..a47c09c 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"] 1780943955 "paper.tex" "paper.pdf" "paper" 1780943957 +["pdflatex"] 1780944443 "paper.tex" "paper.pdf" "paper" 1780944444 "/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 "" @@ -25,14 +25,12 @@ "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti7.tfm" 1136768653 1492 86331993fe614793f5e7e755835c31c5 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 "" - 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zEy>7n;>Pskcrp_zE%DSO<;K8j(yB2lW${1pEhczW6lf-VAXMLjmZiZm$rm8NnoF-^SqH>gl_T> zS|VgGVoS+6VAo1pg*qR!cYBA#dyD?^q||@`o!QIlgf;*6c8g8~ORp2fhW41hJHUn(7?R8cGFV?;h)}rk9GLgHdayBYvmUZaSnD`YzHD995Wje2NMT08JU=ZIQ0Jj{gh51 diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 5f751ee..9f06101 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -28,7 +28,7 @@ \begin{document} -\title{Coloring Nested Tire Graphs} +\title{Nested Tire Decompositions of Plane Triangulations} % author one information \author{Eric Bauerfeld} @@ -46,20 +46,18 @@ \dedicatory{} \begin{abstract} -We establish the foundational structure of nested -level-induced tire decompositions of a plane triangulation $G$. -A \emph{level source} of $G$ induces a BFS layering of $G$ and -endows the inner planar dual $G'$ with a \emph{dual depth} -grading. The basic object of study is the \emph{tire graph} -$T$ --- a plane graph whose outer and inner boundaries bound a -closed planar region, the \emph{tire tread} $R$, triangulated by -the \emph{annular edges} $E_{\mathrm{ann}}$. Our main structural -result, the \emph{tire-component lemma}, exhibits each connected -component of $G'_d$ as a tire graph; the \emph{tire-tread -partition theorem} consequence shows the resulting tire treads -partition the bounded faces of $G$. Coloring questions on -$G$ thereby factor through coloring questions on the -individual treads. +We establish the foundational structure of nested level-induced tire +decompositions of a plane triangulation $G$. A \emph{level source} of +$G$ induces a BFS layering of $G$ and endows the inner planar dual +$G'$ with a \emph{dual depth} grading. The basic object of study is +the \emph{tire graph} $T$ --- a plane graph whose outer and inner +boundaries bound a closed planar region, the \emph{tire tread} $R$, +triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$. We define +medial tire graphs and prove a basic colour-count bound for their +annular medial cycle. Our main structural results are the +\emph{tire-component lemma}, the \emph{tire-tread partition theorem}, +and the rooted \emph{tire-tree decomposition}, which together organize +the bounded faces of $G$ into nested tire treads. \end{abstract} \maketitle @@ -1370,386 +1368,6 @@ This is the structural setup underlying the chain-pigeonhole program for tire treads. \end{remark} -\begin{remark}[Motivation for level-cycle restrictions] -\label{rem:level-cycle-motivation} -The tire-tree decomposition reduces global colouring questions to local -choices on treads together with compatibility along nested boundary -cycles. Without further structure, the number of boundary colour states -can grow quickly as one descends the tree: each seam or level cycle may -in principle carry any proper restriction of a $4$-colouring. The -following restriction is meant to test whether this state space can be -compressed. If level cycles can always be made to omit one colour, then -each such interface behaves like a three-colour boundary object, while -still allowing different cycles to omit different colours. This would -not by itself solve the gluing problem, but it would give a simpler -target class of boundary states for arguments about nested tire trees. -\end{remark} - -\begin{definition}[Level-cycle three-colour restriction] -\label{def:level-cycle-three-colour-restriction} -Let $G$ be a maximal planar graph, let $S \subseteq V(G)$ be a level -source, and let $c \colon V(G) \to \{1,2,3,4\}$ be a proper -$4$-vertex-colouring of $G$. We say that $c$ has the -\emph{level-cycle three-colour restriction} with respect to $S$ if, -for every level $d \geq 0$ and every simple cycle -$C \subseteq G[L_d]$, the colour set used on $C$ has size at most -three: -\[ - |c(V(C))| \leq 3. -\] -Equivalently, every simple cycle contained in a single level omits at -least one of the four colours. The omitted colour may depend on the -cycle; in particular, distinct cycles in the same level, the same tire -tread, or the same inner outerplanar component are not required to -omit the same colour. -\end{definition} - -\begin{conjecture}[False universal-source form] -\label{conj:false-universal-level-cycle-three-colour} -Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be any -level source. Then $G$ admits a proper $4$-vertex-colouring with the -level-cycle three-colour restriction with respect to $S$. -\end{conjecture} - -\begin{figure}[htbp] -\centering -\includegraphics[width=0.78\textwidth]{fig_universal_level_cycle_counterexample.png} -\caption{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.} -\label{fig:universal-level-cycle-counterexample} -\end{figure} - -\begin{example}[Counterexample to Conjecture~\ref{conj:false-universal-level-cycle-three-colour}] -\label{ex:universal-level-cycle-counterexample} -Let $G$ be the maximal planar graph on vertex set -$\{1,2,3,4,5,6,7,8\}$ with edge set -\[ -\begin{aligned} -E(G)=\{& -12,13,14,15,16,17,23,26,27,34,35,36,38,\\ -&45,56,58,67,68\}. -\end{aligned} -\] -Here $ij$ denotes the edge $\{i,j\}$. -Take the vertex source $S=\{7\}$. The corresponding levels are -\[ -L_0=\{7\},\qquad L_1=\{1,2,6\},\qquad -L_2=\{3,4,5,8\}. -\] -Inside $G[L_2]$ the vertices $(3,4,5,8)$ form a simple cycle. In -every proper $4$-vertex-colouring of $G$, these four vertices receive -four distinct colours. The edges $34$, $45$, $58$, $38$, and $35$ -force all pairs among $\{3,4,5,8\}$ except possibly $\{4,8\}$ to have -distinct colours. If $4$ and $8$ had the same colour, then vertex $6$, -which is adjacent to $3$, $5$, and $8$, would have to use the fourth -colour; but vertex $1$ is adjacent to $3$, $4$, $5$, and $6$, and -would then be adjacent to all four colours, impossible in a proper -$4$-colouring. Hence $4$ and $8$ also have distinct colours, so the -level cycle $(3,4,5,8)$ uses all four colours in every proper -$4$-colouring of $G$. 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} - -\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. -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. - -The graph $G^\star$ does not refute -Conjecture~\ref{conj:level-cycle-three-colour}: the vertex source -$v_0 = 10$ admits a proper $4$-colouring under which every simple level -cycle uses at most three colours. - -\subsection*{The surviving level-cycle conjecture} - -\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 -$S \subseteq V(G)$ such that $G$ admits a proper $4$-vertex-colouring -with the level-cycle three-colour restriction with respect to $S$. -\end{conjecture} - -\subsection*{Enumeration for small $n$} - -We exhaustively enumerated all plane triangulation isomorphism classes with -$4 \leq n \leq 13$ 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 -witness. - -\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 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} - -\begin{definition}[Seam] -\label{def:seam} -A \emph{seam} of a maximal planar graph $G$ is a simple cycle -$C \subset G$ such that, for some vertex $v_0 \in V(G)$, $C = -B_{\mathrm{out}}^{(T)}$ for some non-root tread $T$ in -$\mathcal{T}(G, \{v_0\})$. - -By Theorem~\ref{thm:tire-tree-decomposition}, every seam $C$ separates -$G$ into: -\begin{itemize} -\item the \emph{seam interior} $G_T$, the triangulated disk on the - $T$-descendant side of $C$; -\item the \emph{seam exterior} $G_C^{\mathrm{ext}} := G \setminus - \mathrm{int}(G_T)$, the triangulated polygon with outer face - bounded by $C$ on the side containing $v_0$; -\end{itemize} -both sharing $C$. A seam is \emph{non-trivial} if both -$V(G_T) \setminus V(C)$ and $V(G_C^{\mathrm{ext}}) \setminus V(C)$ are -non-empty. - -For any seam $C$ and either side $X \in \{G_T, G_C^{\mathrm{ext}}\}$, -write -\[ - \mathrm{Col}(X \mid C) \;:=\; \bigl\{\, c|_{V(C)} \;:\; c \text{ a - proper $4$-colouring of } X \,\bigr\} \;\subseteq\; \{1,2,3,4\}^{V(C)} -\] -for the set of $C$-restricted $4$-colourings induced by $4$-colourings -of $X$ (each element is a proper $4$-colouring of the cycle $C$). -\end{definition} - -\begin{definition}[Partial tire tree] -\label{def:partial-tire-tree} -Let $T_r$ be a tire tread in $\mathcal{T}(G, S)$ with outer boundary -cycle $C_{T_r} = B_{\mathrm{out}}^{(T_r)}$, and let $G_{T_r}$ be the -triangulated disk inside $C_{T_r}$ given by -Theorem~\ref{thm:tire-tree-decomposition}. The \emph{partial tire -tree} with root $T_r$, written $G_{T_r}^{\circ}$, is the induced -subgraph of $G$ on the vertex set -$V(G_{T_r}) \setminus V(C_{T_r})$ --- -i.e.\ $G_{T_r}$ with the seam-cycle vertices removed. - -Equivalently, $V(G_{T_r}^{\circ})$ is the set of vertices of $G$ -strictly inside $C_{T_r}$ on the side away from the level source, -and $E(G_{T_r}^{\circ})$ consists of the edges of $G$ both of whose -endpoints lie in this strict interior. The tree-of-tire-treads -structure of $G_{T_r}^{\circ}$ is the sub-tree of $\mathcal{T}(G, S)$ -rooted at $T_r$, with $T_r$'s outer boundary peeled away. -\end{definition} - -\begin{lemma}[Seam edges are shared by at most one other depth-$d$ seam] -\label{lem:seam-edge-shared} -Let $G$ be a maximal planar graph with single-vertex level source -$S = \{v_0\}$, fix $d \ge 1$, and let $e \in E(G)$ be an edge lying on -the seam $C_T = B_{\mathrm{out}}^{(T)}$ of some tire tread -$T \in \mathcal{T}(G, S)$ at depth $d$. Then there is at most one -other tire tread $T' \in \mathcal{T}(G, S)$ at the same depth $d$ with -$e \in C_{T'}$. -\end{lemma} - -\begin{proof} -By Theorem~\ref{thm:tread-tree}, $C_T$ is the boundary cycle of a -bounded face of the parent's inner outerplanar graph $O^{(T_p)}$, -where $T_p \in \mathcal{T}(G, S)$ is the parent of $T$ at depth -$d - 1$. The inner dual of a connected outerplanar graph is a tree, -so each edge of -$O^{(T_p)}$ lies on at most two of its bounded face cycles. Hence -$e$ lies on at most one other bounded face cycle of $O^{(T_p)}$, -corresponding (Theorem~\ref{thm:tread-tree}, child--face bijection) -to at most one sibling of $T$ at depth $d$ whose seam contains $e$. -\end{proof} - -\begin{conjecture}[Seam structure of minimum $4$CT counterexamples, sketch] -\label{conj:seam-counterexample} -Suppose the Four Colour Theorem fails: there exists a maximal planar -graph that is not $4$-colourable. Let $G$ be a \emph{minimum} such -counterexample (with $|V(G)|$ minimal among non-$4$-colourable maximal -planar graphs). Then: - -\medskip - -\noindent\emph{Restatement-of-classical content.} -\begin{itemize} -\item[(C1)] \emph{Bilateral colourability.} For every non-trivial seam - $C$ of $G$, both $\mathrm{Col}(G_T \mid C)$ and - $\mathrm{Col}(G_C^{\mathrm{ext}} \mid C)$ are non-empty. -\item[(C2)] \emph{Bilateral incompatibility.} For every non-trivial - seam $C$, - \[ - \mathrm{Col}(G_T \mid C) \;\cap\; - \mathrm{Col}(G_C^{\mathrm{ext}} \mid C) \;=\; \emptyset. - \] -\item[(C3)] \emph{Length lower bound (Birkhoff).} Every non-trivial - seam $C$ of $G$ has $|V(C)| \ge 6$. -\end{itemize} - -(C1) and (C2) together restate ``$G$ is a counterexample whose every -internal cut by a seam splits into two colourable pieces with -incompatible boundary palettes''; (C1) follows from minimality applied -to each side after closing the polygonal outer face by a single apex, -(C2) from $G$ itself being non-$4$-colourable. (C3) is Birkhoff's -internally-$6$-connected condition restated in the seam language. - -\medskip - -\noindent\emph{Substantive (speculative) content.} -\begin{itemize} -\item[(C4)] \emph{Innermost obstruction.} There exists a vertex source - $v_0 \in V(G)$ and a \emph{leaf} tread $T^* \in - \mathcal{T}(G, \{v_0\})$ (a tread with no children in the - tree-of-treads) such that: - \begin{enumerate} - \item[(i)] the seam interior $G_{T^*}$ is, up to plane - iso, one of a finite list of \emph{minimal seam - configurations}, characterized by their boundary - palette $\mathrm{Col}(G_{T^*} \mid C_{T^*})$ being a - specific proper subset of the proper $4$-colourings - of the cycle $C_{T^*}$; - \item[(ii)] the path in $\mathcal{T}(G, \{v_0\})$ from the - root $T_0$ to $T^*$ is an \emph{obstruction chain}: - $\mathrm{Col}(G_T \mid C_T)$ is monotonically - restricted (under the natural pull-back along - parent--child seams of - Remark~\ref{rem:tree-coloring-factorisation}) as $T$ - descends from the root to $T^*$, with the final - restriction at $T^*$ being incompatible with the - $v_0$-side palette. - \end{enumerate} -\end{itemize} -\end{conjecture} - \begin{thebibliography}{9} \bibitem{tait-original} diff --git a/papers/three_color_restrictions_nested_tire_graphs/fig_inner_boundary_counterexample.png b/papers/three_color_restrictions_nested_tire_graphs/fig_inner_boundary_counterexample.png new file mode 100644 index 0000000000000000000000000000000000000000..11da609c489301d7290f19d4ebda8e25ef75fdef GIT binary patch literal 86866 zcmeFZcT|(zw+4un&jxm-i=ZeV2+~_bq$?;&uZl<$Lhm6e0wNs+l#Vo|h29A$O}c;( zAV8#-5PAY6B$<=&-QT^l?wa{`*36o$&(Z+?-rxgoPdcGeQXJVl8}KWxx>3a?DB1(3OFK4MF>Nz>GDTo#6lj zgZh2Nr(Q|R5cUiJ-eL|l;etwkqlaR!_74KTc z`pWuSPG4@4pJ?2BQxYRom1PxsSmWN6u)DuJ&Z5^pSha0L6;QF#AraTKM`U6mGJ8pQ z7Z*1_H@AogH(oe?VK-~XPazd7i2ky;^SyZDzy4r2s`UEQzkd{6JNW9~kFRf?u{iYa z=T{*NZ2x|=&^$c+??(pXtBo{)(O$!64@m5PVrY!}rwBYF{Z(=Q|Bv#RAag>)AJ0Ej|xR=~rJb};=!k1!`bgoAR 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