Define simple and compound medial tires

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@@ -12,33 +12,33 @@
 \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}}
+\citation{bauerfeld-nested-tire-decompositions}
-\newlabel{rem:up-teeth-count}{{3.6}{3}}
+\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}
+\newlabel{def:medial-restriction-relation}{{3.10}{4}}
-\@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 }
+\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{4}{}\protected@file@percent }
 \newlabel{cor:medial-tire-decomposition}{{4.1}{4}}
 \@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 }
+\@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}{5}}
+\newlabel{fig:medial-annular-cycle-counterexample}{{3}{6}}
 \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}}
 \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 }
 \newlabel{lem:kempe-cycles}{{5.5}{7}}
 \newlabel{lem:kempe-conservation}{{5.6}{8}}
 \newlabel{def:kempe-balanced}{{5.7}{8}}
-\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}
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@@ -159,19 +159,27 @@ colours.
 \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
+by~\cite{bauerfeld-nested-tire-decompositions}.  The \emph{medial tire
-tire graph} of $T$, denoted $\mathsf{M}(T)$, is the subgraph of
+graph} of $T$, denoted $\mathsf{M}(T)$, is the subgraph of $M(G)$
-$M(G)$ induced by the medial vertices $m_e$ with $e$ an edge of $G$
+induced by the medial vertices $m_e$ with $e$ an edge of $G$ incident
-incident to at least one triangular face in the tread $T$.  The medial
+to at least one triangular face in the tread $T$.  The medial vertices
-vertices corresponding to annular edges of $T$ are called
+corresponding to annular edges of $T$ are called \emph{annular medial
-\emph{annular medial vertices}.
+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
+decomposition of the full medial graph $M(G)$ into medial tire graphs
-graphs $\{\mathsf{M}(T): T \in V(\mathcal{T}(G,S))\}$, glued along
+$\{\mathsf{M}(T): T \in V(\mathcal{T}(G,S))\}$, glued along their
-their boundary medial vertex sets.
+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