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Define simple and compound medial tires

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2026-06-15 16:14:50 -04:00
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papers/medial_tire_decompositions_of_plane_triangulations/paper.aux
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\newlabel{def:full-medial-tire}{{3.1}{2}}
\newlabel{thm:annular-medial-colour-bound}{{3.3}{3}}
\newlabel{def:annular-teeth}{{3.4}{3}}
\newlabel{rem:teeth-sharing}{{3.5}{3}}
\newlabel{rem:up-teeth-count}{{3.6}{3}}
\citation{bauerfeld-nested-tire-decompositions}
\citation{bauerfeld-nested-tire-decompositions}
\newlabel{rem:teeth-sharing}{{3.5}{4}}
\newlabel{rem:up-teeth-count}{{3.6}{4}}
\newlabel{def:bite}{{3.7}{4}}
\newlabel{rem:bite-face-count}{{3.8}{4}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A full medial tire graph $\mathsf {M}(T)$ illustrating the tooth terminology. The thick cycle is the annular medial cycle $A(T)$, whose black vertices are the annular medial vertices. Each edge of $A(T)$ carries one tooth: up teeth (blue apexes, outer-boundary medial vertices) point into the outer region, and down teeth (red apexes, inner-boundary medial vertices) point into the inner region. The two down teeth meeting at the central shared apex (larger red vertex) form a bite; that shared apex splits the inner region into two faces, one with four down teeth on its boundary and one with none.}}{4}{}\protected@file@percent }
\newlabel{fig:medial-teeth-example}{{1}{4}}
\newlabel{def:boundary-medial-vertices}{{3.9}{4}}
\citation{bauerfeld-nested-tire-decompositions}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Three six-face full medial tire graphs found by the boundary-state restriction search. Black vertices are annular medial vertices; blue vertices are outer boundary medial vertices and red vertices are inner boundary medial vertices. The word below each diagram records the outer/inner type of the six annular faces in cyclic order. Boundary states are identified only up to colour permutation, not by rotation or reflection of the boundary order.}}{5}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A simple medial tire graph $\mathsf {M}(T)$ illustrating the tooth terminology. The thick cycle is the annular medial cycle $A(T)$, whose black vertices are the annular medial vertices. Each edge of $A(T)$ carries one tooth: up teeth (blue apexes, outer-boundary medial vertices) point into the outer region, and down teeth (red apexes, inner-boundary medial vertices) point into the inner region. The two down teeth meeting at the central shared apex (larger red vertex) form a bite; that shared apex splits the inner region into two faces, one with four down teeth on its boundary and one with none.}}{5}{}\protected@file@percent }
\newlabel{fig:medial-teeth-example}{{1}{5}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Three six-face simple medial tire graphs found by the boundary-state restriction search. Black vertices are annular medial vertices; blue vertices are outer boundary medial vertices and red vertices are inner boundary medial vertices. The word below each diagram records the outer/inner type of the six annular faces in cyclic order. Boundary states are identified only up to colour permutation, not by rotation or reflection of the boundary order.}}{5}{}\protected@file@percent }
\newlabel{fig:medial-restriction-worst-cases}{{2}{5}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A proper vertex $3$-colouring of the full medial graph of the first seven-vertex counterexample found by the experiment. The medial vertex labelled $ij$ corresponds to the edge $(i,j)$ of the triangulation. For the vertex-source decomposition at source $1$, the highlighted annular medial cycle has colour counts $(2,2,2)$, so it is not coloured with two colours except at at most one vertex.}}{5}{}\protected@file@percent }
\newlabel{fig:medial-annular-cycle-counterexample}{{3}{5}}
\newlabel{def:medial-restriction-relation}{{3.10}{5}}
\citation{bauerfeld-nested-tire-decompositions}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{6}{}\protected@file@percent }
\newlabel{cor:medial-tire-decomposition}{{4.1}{6}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A proper vertex $3$-colouring of the full medial graph of the first seven-vertex counterexample found by the experiment. The medial vertex labelled $ij$ corresponds to the edge $(i,j)$ of the triangulation. For the vertex-source decomposition at source $1$, the highlighted annular medial cycle has colour counts $(2,2,2)$, so it is not coloured with two colours except at at most one vertex.}}{6}{}\protected@file@percent }
\newlabel{fig:medial-annular-cycle-counterexample}{{3}{6}}
\newlabel{def:compatible-family}{{4.2}{6}}
\newlabel{prop:gluing-criterion}{{4.3}{6}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{6}{}\protected@file@percent }
\newlabel{def:medial-boundary-state}{{5.1}{6}}
\newlabel{def:medial-boundary-state}{{5.1}{7}}
\newlabel{conj:medial-chain-pigeonhole}{{5.2}{7}}
\newlabel{conj:medial-route-fct}{{5.3}{7}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Kempe-cycle conservation across medial tires}}{7}{}\protected@file@percent }
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\newlabel{rem:kempe-balance-necessary}{{5.8}{8}}
\newlabel{rem:kempe-balance-necessary}{{5.8}{9}}
\bibcite{bauerfeld-nested-tire-decompositions}{1}
\bibcite{tait-original}{2}
\newlabel{tocindent-1}{0pt}
papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk
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papers/medial_tire_decompositions_of_plane_triangulations/paper.log
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papers/medial_tire_decompositions_of_plane_triangulations/paper.tex
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\section{Medial tire pieces}
\begin{definition}[Full medial tire graph]
\begin{definition}[Simple and compound medial tire graphs]
\label{def:full-medial-tire}
Let $T$ be a tire tread in the tire tree $\mathcal{T}(G,S)$ supplied
by~\cite{bauerfeld-nested-tire-decompositions}. The \emph{full medial
tire graph} of $T$, denoted $\mathsf{M}(T)$, is the subgraph of
$M(G)$ induced by the medial vertices $m_e$ with $e$ an edge of $G$
incident to at least one triangular face in the tread $T$. The medial
vertices corresponding to annular edges of $T$ are called
\emph{annular medial vertices}.
by~\cite{bauerfeld-nested-tire-decompositions}. The \emph{medial tire
graph} of $T$, denoted $\mathsf{M}(T)$, is the subgraph of $M(G)$
induced by the medial vertices $m_e$ with $e$ an edge of $G$ incident
to at least one triangular face in the tread $T$. The medial vertices
corresponding to annular edges of $T$ are called \emph{annular medial
vertices}.
We call $\mathsf{M}(T)$ a \emph{simple medial tire graph} if its
annular medial vertices induce a single cycle. We call
$\mathsf{M}(T)$ a \emph{compound medial tire graph} if it is associated
to a connected depth component of tread faces but its annular medial
vertices induce more than one cycle. In a compound medial tire graph,
annular teeth are understood cycle-by-cycle, and up-tooth apexes
belonging to different annular cycles may coincide.
\end{definition}
\begin{remark}
In the ambient-triangulation setting, the full medial tire graph
In the ambient-triangulation setting, the simple medial tire graph
$\mathsf{M}(T)$ coincides with the omitted-edge medial tire graph
studied in~\cite{bauerfeld-nested-tire-decompositions}. Indeed, the
medial edges of $\mathsf{M}(T)$ are contributed by corners of annular
@@ -361,7 +369,7 @@ interior.
\node[lbl, anchor=east] at (192:1.86) (L0) {region with 0 down teeth};
\draw[lead] (L0.east) -- (180:0.45);
\end{tikzpicture}
\caption{A full medial tire graph $\mathsf{M}(T)$ illustrating the tooth
\caption{A simple medial tire graph $\mathsf{M}(T)$ illustrating the tooth
terminology. The thick cycle is the annular medial cycle $A(T)$, whose
black vertices are the annular medial vertices. Each edge of $A(T)$
carries one tooth: up teeth (blue apexes, outer-boundary medial vertices)
@@ -453,7 +461,7 @@ its boundary and one with none.}
\node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$};
\end{scope}
\end{tikzpicture}
\caption{Three six-face full medial tire graphs found by the boundary-state
\caption{Three six-face simple medial tire graphs found by the boundary-state
restriction search. Black vertices are annular medial vertices; blue
vertices are outer boundary medial vertices and red vertices are inner
boundary medial vertices. The word below each diagram records the
@@ -601,9 +609,11 @@ parent and children.
Let $G$ be a plane triangulation with level source $S$. The tire-tree
decomposition $\mathcal{T}(G,S)$ of
\cite{bauerfeld-nested-tire-decompositions} induces a rooted
decomposition of the full medial graph $M(G)$ into full medial tire
graphs $\{\mathsf{M}(T): T \in V(\mathcal{T}(G,S))\}$, glued along
their boundary medial vertex sets.
decomposition of the full medial graph $M(G)$ into medial tire graphs
$\{\mathsf{M}(T): T \in V(\mathcal{T}(G,S))\}$, glued along their
boundary medial vertex sets. A node of this decomposition may be a
simple medial tire graph or a compound medial tire graph, depending on
whether its annular medial vertices induce one cycle or several.
\end{corollary}
\begin{proof}
@@ -670,6 +680,17 @@ this framework would follow from a structural reason that these
restriction sets cannot remain mutually disjoint along every branch of
the tire tree.
In this chaining step, the inner side of a parent simple medial tire is
read by its singleton down-tooth apex vertices. If the child side is a
compound medial tire, then the parent's singleton down-tooth apex
vertices are incident to---indeed, are identified with---the up-tooth
apex vertices of the compound medial tire, interpreted cycle-by-cycle
on its annular medial cycles. Equivalently, the primal edges
represented by the parent's singleton down-tooth apexes are exactly the
level-cycle interface edges represented on the child side as up-tooth
apexes. This is the boundary identification along which the medial
boundary states are chained.
\begin{definition}[Medial boundary state]
\label{def:medial-boundary-state}
A \emph{medial boundary state} on a boundary set
@@ -786,12 +807,12 @@ $C$. Thus every entrance through $C$ is paired with an exit through
$C$.
\end{proof}
We now use these Kempe cycles to single out the colourings of a full
We now use these Kempe cycles to single out the colourings of a simple
medial tire graph that respect the annular tooth structure.
\begin{definition}[Kempe-balanced colouring]
\label{def:kempe-balanced}
Let $\varphi$ be a proper $3$-colouring of the full medial tire graph
Let $\varphi$ be a proper $3$-colouring of the simple medial tire graph
$\mathsf{M}(T)$. For a colour pair $P=\{a,b\}$, let $\mathsf{M}(T)_P$ be
the subgraph induced by the vertices of colours $a$ and $b$. Since
$\mathsf{M}(T)$ need not be $4$-regular, the components of