Split three-colour restrictions into separate paper
This commit is contained in:
@@ -8,15 +8,15 @@
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\citation{dvorak-lidicky-cones}
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\citation{heesch-untersuchungen}
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\citation{robertson-sanders-seymour-thomas}
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\citation{robertson-sanders-seymour-thomas}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent }
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\citation{robertson-sanders-seymour-thomas}
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\newlabel{def:dual}{{1.3}{2}}
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\newlabel{def:dual-depth}{{1.4}{2}}
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\newlabel{def:dual-component}{{1.5}{2}}
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\newlabel{def:tire-graph}{{1.6}{2}}
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\@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 }
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\newlabel{fig:dual-depth}{{1}{3}}
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\newlabel{def:tire-graph}{{1.6}{3}}
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\@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 }
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\newlabel{fig:tire-example}{{2}{4}}
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\newlabel{def:medial-tire-graph}{{1.7}{4}}
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@@ -49,34 +49,6 @@
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\newlabel{thm:tread-tree}{{1.23}{14}}
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\newlabel{rem:tree-multiple-children}{{1.24}{15}}
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\newlabel{thm:tire-tree-decomposition}{{1.25}{15}}
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\newlabel{rem:tree-coloring-factorisation}{{1.26}{17}}
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\newlabel{rem:level-cycle-motivation}{{1.27}{17}}
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\newlabel{def:level-cycle-three-colour-restriction}{{1.28}{17}}
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\@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 }
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\newlabel{fig:tire-tree-decomposition}{{6}{18}}
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\newlabel{conj:false-universal-level-cycle-three-colour}{{1.29}{18}}
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\newlabel{ex:universal-level-cycle-counterexample}{{1.30}{18}}
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\@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 }
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\newlabel{fig:universal-level-cycle-counterexample}{{7}{19}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{An inner-boundary refinement}}{19}{}\protected@file@percent }
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\newlabel{def:tire-inner-boundary-three-colour}{{1.31}{19}}
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\newlabel{conj:tire-inner-boundary-three-colour}{{1.32}{19}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A counterexample at $n=14$}}{20}{}\protected@file@percent }
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\newlabel{ex:inner-boundary-counterexample}{{1.33}{20}}
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\@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 }
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\newlabel{fig:inner-boundary-counterexample}{{8}{20}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{21}{}\protected@file@percent }
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\newlabel{conj:level-cycle-three-colour}{{1.34}{21}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{21}{}\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.}}{21}{}\protected@file@percent }
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\newlabel{tab:level-cycle-three-colour-counts}{{1}{21}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{21}{}\protected@file@percent }
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\newlabel{def:seam}{{1.35}{21}}
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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.}}{22}{}\protected@file@percent }
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\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{22}}
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\newlabel{def:partial-tire-tree}{{1.36}{22}}
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\newlabel{lem:seam-edge-shared}{{1.37}{22}}
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\newlabel{conj:seam-counterexample}{{1.38}{22}}
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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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@@ -93,5 +65,8 @@
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\newlabel{tocindent1}{17.77782pt}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{23}{}\protected@file@percent }
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\gdef \@abspage@last{23}
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\newlabel{rem:tree-coloring-factorisation}{{1.26}{17}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{17}{}\protected@file@percent }
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\@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 }
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\newlabel{fig:tire-tree-decomposition}{{6}{18}}
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\gdef \@abspage@last{18}
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@@ -1,5 +1,5 @@
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@@ -477,16 +477,6 @@ INPUT ./fig_tire_tree_decomposition.png
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INPUT ./fig_tire_tree_decomposition.png
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[9] [10] [11] [12] [13] [14] [15] [16]
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[]\OT1/cmr/m/n/10 Length lower bound (Birkhoff). \OT1/cmr/m/it/10 Ev-ery non-tr
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[]
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@@ -28,7 +28,7 @@
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\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}
|
||||
|
||||
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|
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@@ -0,0 +1,64 @@
|
||||
\relax
|
||||
\citation{bauerfeld-nested-tire-decompositions}
|
||||
\citation{bauerfeld-nested-tire-duals}
|
||||
\citation{birkhoff-reducibility}
|
||||
\citation{birkhoff-lewis-chromatic}
|
||||
\citation{tutte-chromatic-sums-1973}
|
||||
\citation{tutte-algebraic-colorings}
|
||||
\citation{tutte-four-colour-conjecture}
|
||||
\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{bauerfeld-nested-tire-decompositions}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Background from nested tire decompositions}}{2}{}\protected@file@percent }
|
||||
\newlabel{rem:level-cycle-motivation}{{2.1}{2}}
|
||||
\newlabel{def:level-cycle-three-colour-restriction}{{2.2}{2}}
|
||||
\newlabel{conj:false-universal-level-cycle-three-colour}{{2.3}{2}}
|
||||
\newlabel{ex:universal-level-cycle-counterexample}{{2.4}{2}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\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.}}{3}{}\protected@file@percent }
|
||||
\newlabel{fig:universal-level-cycle-counterexample}{{1}{3}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{An inner-boundary refinement}}{3}{}\protected@file@percent }
|
||||
\newlabel{def:tire-inner-boundary-three-colour}{{2.5}{3}}
|
||||
\newlabel{conj:tire-inner-boundary-three-colour}{{2.6}{4}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A counterexample at $n=14$}}{4}{}\protected@file@percent }
|
||||
\newlabel{ex:inner-boundary-counterexample}{{2.7}{4}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The $14$-vertex counterexample $G^\star $ to Conjecture\nonbreakingspace 2.6\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 $12$.}}{4}{}\protected@file@percent }
|
||||
\newlabel{fig:inner-boundary-counterexample}{{2}{4}}
|
||||
\citation{bauerfeld-nested-tire-decompositions}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{5}{}\protected@file@percent }
|
||||
\newlabel{conj:level-cycle-three-colour}{{2.8}{5}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{5}{}\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.}}{5}{}\protected@file@percent }
|
||||
\newlabel{tab:level-cycle-three-colour-counts}{{1}{5}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{5}{}\protected@file@percent }
|
||||
\newlabel{def:seam}{{2.9}{5}}
|
||||
\citation{bauerfeld-nested-tire-decompositions}
|
||||
\citation{bauerfeld-nested-tire-decompositions}
|
||||
\@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.}}{6}{}\protected@file@percent }
|
||||
\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{6}}
|
||||
\newlabel{def:partial-tire-tree}{{2.10}{6}}
|
||||
\newlabel{lem:seam-edge-shared}{{2.11}{6}}
|
||||
\citation{bauerfeld-nested-tire-decompositions}
|
||||
\bibcite{tait-original}{1}
|
||||
\bibcite{bauerfeld-depth}{2}
|
||||
\bibcite{bauerfeld-nested-tire-decompositions}{3}
|
||||
\bibcite{bauerfeld-nested-tire-duals}{4}
|
||||
\bibcite{birkhoff-reducibility}{5}
|
||||
\bibcite{birkhoff-lewis-chromatic}{6}
|
||||
\bibcite{tutte-four-colour-conjecture}{7}
|
||||
\bibcite{tutte-algebraic-colorings}{8}
|
||||
\bibcite{tutte-chromatic-sums-1973}{9}
|
||||
\bibcite{heesch-untersuchungen}{10}
|
||||
\bibcite{robertson-sanders-seymour-thomas}{11}
|
||||
\bibcite{dvorak-lidicky-cones}{12}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{14.69437pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\newlabel{conj:seam-counterexample}{{2.12}{7}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{8}
|
||||
@@ -0,0 +1,145 @@
|
||||
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||||
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%% filename: amsart-template.tex
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%% American Mathematical Society
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%% AMS-LaTeX v.2 template for use with amsart
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%% ====================================================================
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||||
\documentclass{amsart}
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\usepackage{amssymb}
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\usepackage{graphicx}
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\usepackage{tikz}
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\usetikzlibrary{backgrounds}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{corollary}[theorem]{Corollary}
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\newtheorem{proposition}[theorem]{Proposition}
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||||
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||||
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||||
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||||
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\numberwithin{equation}{section}
|
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|
||||
\begin{document}
|
||||
|
||||
\title{Three-Colour Restrictions for Nested Tire Graphs}
|
||||
|
||||
% author one information
|
||||
\author{Eric Bauerfeld}
|
||||
\address{}
|
||||
\curraddr{}
|
||||
\email{}
|
||||
\thanks{}
|
||||
|
||||
\subjclass[2010]{Primary }
|
||||
|
||||
\keywords{plane graph, triangulation, plane depth, level edge, dual graph, tire graph}
|
||||
|
||||
\date{}
|
||||
|
||||
\dedicatory{}
|
||||
|
||||
\begin{abstract}
|
||||
We study three-colour boundary restrictions suggested by the nested
|
||||
tire decomposition of a plane triangulation. A level source induces a
|
||||
rooted tree of tire treads, and global colouring questions factor
|
||||
through local tread colourings together with compatibility along nested
|
||||
boundary cycles. We formulate a level-cycle three-colour restriction,
|
||||
exhibit counterexamples to two overly strong forms, and record
|
||||
exhaustive evidence for a surviving source-dependent conjecture. We
|
||||
also introduce seam language for minimum Four Colour Theorem
|
||||
counterexamples in the tire-tree framework.
|
||||
\end{abstract}
|
||||
|
||||
\maketitle
|
||||
|
||||
\section{Introduction}
|
||||
|
||||
A classical theorem of Tait recasts the Four Colour Theorem in dual,
|
||||
edge-colouring terms: a plane triangulation $G$ is properly $4$-vertex-colourable
|
||||
if and only if its dual cubic graph $G'$ is properly $3$-edge-colourable. Thus a
|
||||
minimal counterexample to the Four Colour Theorem -- a smallest triangulation
|
||||
admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph
|
||||
admitting no proper $3$-edge-colouring.
|
||||
|
||||
The structural study of such a minimal counterexample is the
|
||||
overarching motivation for the present line of work. The companion
|
||||
decomposition paper~\cite{bauerfeld-nested-tire-decompositions}
|
||||
establishes the foundational vocabulary --- level sources, dual depth,
|
||||
tire graphs, medial tire graphs, and tire-tree decompositions --- on
|
||||
which this paper builds. The companion dual paper
|
||||
\cite{bauerfeld-nested-tire-duals} develops nested-cycle structure
|
||||
theorems and chain-pigeonhole conjectures for tire annular subgraphs
|
||||
of $G'$.
|
||||
|
||||
\paragraph{Related work.}
|
||||
The structural object underlying this programme --- the set of
|
||||
proper $4$-colourings of a boundary cycle that extend to a colouring
|
||||
of a bounded planar region --- is classical. Birkhoff's reducibility
|
||||
analysis of the diamond configuration~\cite{birkhoff-reducibility} is
|
||||
the earliest instance of computing such extension sets to attack the
|
||||
Four Colour Theorem; the chromatic polynomial framework of Birkhoff
|
||||
and Lewis~\cite{birkhoff-lewis-chromatic} systematised the counting.
|
||||
Tutte studied how the chromatic polynomial of a rooted planar
|
||||
triangulation decomposes along its outer
|
||||
boundary~\cite{tutte-chromatic-sums-1973} and developed an algebraic
|
||||
theory of graph colourings organised around separating
|
||||
subgraphs~\cite{tutte-algebraic-colorings, tutte-four-colour-conjecture}.
|
||||
The most recent and structurally closest parallel is Dvo\v{r}\'ak
|
||||
and Lidick\'y's analysis of \emph{coloring count
|
||||
cones}~\cite{dvorak-lidicky-cones}, which characterises the possible
|
||||
boundary-extension functions on a fixed outer cycle of a
|
||||
near-triangulation. The Heesch--Appel--Haken
|
||||
approach~\cite{heesch-untersuchungen, robertson-sanders-seymour-thomas}
|
||||
also uses boundary-extension reasoning, but case-by-case on a finite
|
||||
unavoidable set of local configurations rather than as part of a
|
||||
global structural induction.
|
||||
|
||||
The tire-tree decomposition used here differs from each of
|
||||
these in shape rather than ingredients. Birkhoff, Tutte, and
|
||||
Dvo\v{r}\'ak--Lidick\'y all study \emph{one} boundary; Heesch and
|
||||
the cleaned-up Appel--Haken proof~\cite{robertson-sanders-seymour-thomas}
|
||||
study a finite collection of local boundaries. The present framework
|
||||
organises the entire triangulation into a hierarchy of annular
|
||||
regions glued along level cycles, and asks whether boundary-extension
|
||||
constraints compose compatibly up the hierarchy. To the authors'
|
||||
knowledge, no prior work on the Four Colour Theorem has been
|
||||
organised around a global nested-cycle decomposition of this kind.
|
||||
|
||||
\section{Background from nested tire decompositions}
|
||||
|
||||
We use the terminology and structural results of~\cite{bauerfeld-nested-tire-decompositions}. In particular, a level source induces levels in a plane maximal planar graph, the depth-$d$ inner-dual components determine tire graphs, and the resulting tire treads form a rooted tire tree $\mathcal{T}(G,S)$. For a tread $T$, we write $B_{\mathrm{out}}^{(T)}$ and $B_{\mathrm{in}}^{(T)}$ for its outer and inner boundary data, $O^{(T)}$ for its inner outerplanar graph, and $G_T$ for the triangulated disk on the descendant side of $B_{\mathrm{out}}^{(T)}$. The base paper also records the boundary-state transfer viewpoint for a single tire and the factorisation of global colouring questions through local tread colourings together with compatibility along parent-child interfaces.
|
||||
|
||||
\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, 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 the tire-tree decomposition theorem of
|
||||
\cite{bauerfeld-nested-tire-decompositions}, 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 the tire-tree
|
||||
decomposition theorem of~\cite{bauerfeld-nested-tire-decompositions}.
|
||||
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 the child--face correspondence in the tire-tree construction of
|
||||
\cite{bauerfeld-nested-tire-decompositions}, $C_T$ 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, by the same 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
|
||||
the parent--child seam pull-back described in
|
||||
\cite{bauerfeld-nested-tire-decompositions}) 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}
|
||||
P.~G.~Tait,
|
||||
\emph{Remarks on the colouring of maps},
|
||||
Proc.\ Roy.\ Soc.\ Edinburgh \textbf{10} (1880), 729.
|
||||
|
||||
\bibitem{bauerfeld-depth}
|
||||
E.~Bauerfeld,
|
||||
\emph{Plane Depth},
|
||||
manuscript (math-research repository), 2026.
|
||||
|
||||
\bibitem{bauerfeld-nested-tire-decompositions}
|
||||
E.~Bauerfeld,
|
||||
\emph{Nested Tire Decompositions of Plane Triangulations},
|
||||
manuscript (math-research repository), 2026.
|
||||
|
||||
\bibitem{bauerfeld-nested-tire-duals}
|
||||
E.~Bauerfeld,
|
||||
\emph{Coloring Nested Tire Dual Graphs},
|
||||
manuscript (math-research repository), 2026.
|
||||
|
||||
\bibitem{birkhoff-reducibility}
|
||||
G.~D.~Birkhoff,
|
||||
\emph{The reducibility of maps},
|
||||
Amer.\ J.\ Math.\ \textbf{35} (1913), 115--128.
|
||||
|
||||
\bibitem{birkhoff-lewis-chromatic}
|
||||
G.~D.~Birkhoff and D.~C.~Lewis,
|
||||
\emph{Chromatic polynomials},
|
||||
Trans.\ Amer.\ Math.\ Soc.\ \textbf{60} (1946), 355--451.
|
||||
|
||||
\bibitem{tutte-four-colour-conjecture}
|
||||
W.~T.~Tutte,
|
||||
\emph{On the four-colour conjecture},
|
||||
Proc.\ London Math.\ Soc.\ (2) \textbf{50} (1948), 137--149.
|
||||
|
||||
\bibitem{tutte-algebraic-colorings}
|
||||
W.~T.~Tutte,
|
||||
\emph{On the algebraic theory of graph colorings},
|
||||
J.\ Combin.\ Theory \textbf{1} (1966), 15--50.
|
||||
|
||||
\bibitem{tutte-chromatic-sums-1973}
|
||||
W.~T.~Tutte,
|
||||
\emph{Chromatic sums for rooted planar triangulations: the cases $\lambda = 1$ and $\lambda = 2$},
|
||||
Canad.\ J.\ Math.\ \textbf{25} (1973), 426--447.
|
||||
|
||||
\bibitem{heesch-untersuchungen}
|
||||
H.~Heesch,
|
||||
\emph{Untersuchungen zum Vierfarbenproblem},
|
||||
Hochschulskriptum 810/a/b, Bibliographisches Institut, Mannheim, 1969.
|
||||
|
||||
\bibitem{robertson-sanders-seymour-thomas}
|
||||
N.~Robertson, D.~P.~Sanders, P.~D.~Seymour, and R.~Thomas,
|
||||
\emph{The four-colour theorem},
|
||||
J.\ Combin.\ Theory Ser.\ B \textbf{70} (1997), 2--44.
|
||||
|
||||
\bibitem{dvorak-lidicky-cones}
|
||||
Z.~Dvo\v{r}\'ak and B.~Lidick\'y,
|
||||
\emph{Coloring count cones of planar graphs},
|
||||
J.\ Graph Theory \textbf{100} (2022), 84--100.
|
||||
|
||||
\end{thebibliography}
|
||||
|
||||
\end{document}
|
||||
Reference in New Issue
Block a user