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\usepackage{tikz}
\usetikzlibrary{backgrounds}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}

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\newtheorem{definition}[theorem]{Definition}
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\newtheorem{remark}[theorem]{Remark}

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\begin{document}

\title{Medial Tire Decompositions of Plane Triangulations}

%    author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}

\subjclass[2010]{Primary }

\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Four Colour Theorem}

\date{}

\dedicatory{}

\begin{abstract}
We use the nested tire decomposition of a plane triangulation to induce
a decomposition of its full medial graph into medial tire subgraphs.
For a plane triangulation $G$, the medial graph $M(G)$ is naturally
isomorphic to the medial graph of the planar dual $G^*$, and proper
$3$-vertex-colourings of $M(G)$ are equivalent to proper
$3$-edge-colourings of the cubic dual.  Thus Tait's reformulation of
the Four Colour Theorem may be studied through proper vertex
$3$-colourings of medial subgraphs.  We define medial tire pieces,
their boundary-state restriction relations, and a chain-pigeonhole
conjecture for compatible medial boundary states across the tire tree.
\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.  The present paper records a medial
version of this viewpoint.  The vertices of the medial graph $M(G)$
correspond to edges of $G$, and adjacency in $M(G)$ records
consecutiveness of edges around vertices and faces of $G$.  Since
planar duality interchanges vertices and faces while preserving the
edge set, $M(G)$ is naturally isomorphic to $M(G^*)$.

Consequently a proper vertex $3$-colouring of $M(G)$ is the same
object as a proper edge $3$-colouring of $G^*$.  This suggests another
route toward the Four Colour Theorem: rather than colouring the dual
cubic graph directly, decompose the full medial graph into local
annular pieces and try to prove that their proper vertex
$3$-colouring boundary restrictions always compose.

The structural input is the nested tire decomposition of
\cite{bauerfeld-nested-tire-decompositions}.  A level source in a plane
triangulation determines a rooted tree of tire treads.  Each tread is
an annular triangulated region with an outer boundary, an inner
outerplanar graph, and annular triangular faces.  We show that this
decomposition induces a decomposition of $M(G)$ into medial tire
subgraphs.  The boundary data of a medial tire are proper
$3$-colourings of the medial vertices corresponding to boundary edges
in the associated dual tire graph.

\section{Background}

Throughout, $G$ is a simple plane maximal planar graph with fixed
embedding, and $G^*$ denotes its full planar dual.  We use the level
source, dual depth, tire graph, tire tread, and tire-tree terminology
of~\cite{bauerfeld-nested-tire-decompositions}.  In particular, a level
source $S$ determines a rooted tire tree $\mathcal{T}(G,S)$ whose
vertices are tire treads and whose parent-child relation records
nested containment across level-cycle interfaces.

\begin{definition}[Medial graph]
\label{def:medial-graph}
Let $H$ be a plane graph.  The \emph{medial graph} $M(H)$ has one
vertex $m_e$ for each edge $e \in E(H)$.  Two medial vertices
$m_e,m_f$ are adjacent whenever $e$ and $f$ are consecutive in the
cyclic order of edges around a vertex of $H$ or around a face of $H$.
The embedding is the standard one obtained by placing $m_e$ at the
midpoint of $e$ and drawing medial edges through the vertex- and
face-corners of $H$.
\end{definition}

\begin{remark}
If $H$ has bridges or vertices of degree $1$, the usual medial
construction may create parallel edges or loops depending on the
chosen convention.  In this paper the main application is to plane
triangulations and their cubic planar duals, where the medial graph is
a loopless $4$-regular plane graph.
\end{remark}

\begin{proposition}[Medial dual invariance]
\label{prop:medial-dual-invariance}
Let $H$ be a connected plane graph and let $H^*$ be its planar dual.
Then there is a natural plane-graph isomorphism
\[
    M(H) \cong M(H^*).
\]
\end{proposition}

\begin{proof}
Each edge $e \in E(H)$ corresponds to a unique dual edge $e^* \in
E(H^*)$, giving a bijection $m_e \mapsto m_{e^*}$ between the vertices
of $M(H)$ and $M(H^*)$.  In $M(H)$ two vertices $m_e,m_f$ are adjacent
exactly when $e$ and $f$ are consecutive around either a vertex or a
face of $H$.  Under duality, vertices and faces are interchanged, and
the cyclic order of the corresponding dual edges around the dual face
or dual vertex is the same up to reversal.  Thus the same pairs are
medial-adjacent in $M(H^*)$, and the midpoint construction identifies
the two embedded medial graphs.
\end{proof}

\begin{corollary}[Tait colourings as medial vertex colourings]
\label{cor:tait-medial}
Let $G$ be a simple plane triangulation.  Proper vertex
$3$-colourings of $M(G)$ are in natural bijection with proper
$3$-edge-colourings of the cubic planar dual $G^*$.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:medial-dual-invariance}, $M(G) \cong
M(G^*)$.  Vertices of $M(G^*)$ correspond to edges of $G^*$, and two
such vertices are adjacent exactly when the corresponding dual edges
are incident and consecutive around a vertex or face of $G^*$.  Since
$G^*$ is cubic, proper vertex $3$-colouring of $M(G^*)$ is therefore
equivalent to assigning three colours to the edges of $G^*$ so that the
three edges incident to each dual vertex receive pairwise distinct
colours.
\end{proof}

\section{Medial tire pieces}

\begin{definition}[Full medial tire graph]
\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}.
\end{definition}

\begin{remark}
In the ambient-triangulation setting, the full 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
triangular tread faces.  Such a face contains at most one outer-boundary
edge and at most one inner-boundary edge, so it does not contribute a
medial edge between two outer-boundary edges or between two
inner-boundary edges.  Similarly, chords of the inner outerplanar graph
lie outside the annular tread and are not incident to annular tread
faces.  Thus the deletion rule used for the earlier reduced medial tire
graph removes no edges from the ambient object $\mathsf{M}(T)$.

The distinction only appears in the standalone drawing convention where
the outer and inner boundary walks are added as artificial faces before
forming a medial graph.  Those artificial faces create same-boundary
medial edges, and the reduced construction deletes them.
\end{remark}

\begin{theorem}[Annular medial colour bound]
\label{thm:annular-medial-colour-bound}
Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire tread with
non-degenerate boundaries and simple inner boundary $B_{\mathrm{in}}$.
Let $A(T)$ be the subgraph of $\mathsf{M}(T)$ induced by the annular
medial vertices.  For a graph $H$, write $\operatorname{Col}_3(H)$ for
the set of proper $3$-vertex-colourings of $H$.  Then $A(T)$ is a
cycle---the \emph{annular cycle} of $T$---and
\[
    |\operatorname{Col}_3(\mathsf{M}(T))|
    \;\leq\; |\operatorname{Col}_3(A(T))|.
\]
\end{theorem}

\begin{proof}
Since the tread is a triangulated annulus with no vertices in its
interior, each annular face has exactly one boundary edge, lying either
on $B_{\mathrm{out}}$ or on $B_{\mathrm{in}}$, and exactly two annular
edges.  As the annular faces are traversed cyclically around the tread,
consecutive faces share one annular edge.  Equivalently, the annular
edges occur in a cyclic order in which each annular face contains two
consecutive annular edges.  Hence the subgraph of $\mathsf{M}(T)$
induced by the annular medial vertices is a cycle.

Consider the restriction map from proper $3$-colourings of
$\mathsf{M}(T)$ to colourings of this annular medial cycle $A(T)$.  We
claim that this map is injective.  Let $x$ be a non-annular medial
vertex.  Then $x$ corresponds to an edge of $B_{\mathrm{out}}$ or
$B_{\mathrm{in}}$: chords of $O$ are not incident to annular tread
faces, and hence do not contribute vertices of $\mathsf{M}(T)$.  This
boundary edge is incident to a unique annular face of the tread, and
the other two edges of that face are annular edges.  Therefore $x$ is
adjacent in $\mathsf{M}(T)$ to the two annular medial vertices
corresponding to those two annular edges.

Those two annular medial vertices are adjacent to each other, because
their annular edges are consecutive on the same triangular annular
face.  In any proper $3$-colouring they therefore receive two distinct
colours, and $x$ is forced to receive the remaining third colour.  Thus
every non-annular medial vertex has its colour uniquely determined by
the colouring of $A(T)$.  Two colourings of $\mathsf{M}(T)$ with the
same restriction to $A(T)$ are identical, so the restriction map is
injective.  The stated inequality follows.
\end{proof}

\begin{definition}[Annular teeth]
\label{def:annular-teeth}
By Theorem~\ref{thm:annular-medial-colour-bound} the annular medial
vertices induce the cycle $A(T)$ in $\mathsf{M}(T)$, the annular cycle, so
the edges of $\mathsf{M}(T)$ joining two annular medial vertices are
exactly the edges of $A(T)$.  Each such edge lies in exactly one triangle ($3$-cycle) of
$\mathsf{M}(T)$, and the third vertex of that triangle is necessarily
non-annular, since $A(T)$ has no chords.  We call this triangle an
\emph{annular tooth} and its non-annular vertex the \emph{apex} of the
tooth.

The cycle $A(T)$ separates the plane into two regions: the \emph{outer
region}, which contains the outer-boundary medial vertices, and the
\emph{inner region}, which contains the inner-boundary medial vertices.
An annular tooth is an \emph{up tooth} if its apex lies in the outer
region, and a \emph{down tooth} if its apex lies in the inner region.
\end{definition}

\begin{remark}
\label{rem:teeth-sharing}
The apexes of annular teeth satisfy two sharing bounds: no two up teeth
share an apex, and at most two down teeth share an apex.
\end{remark}

\begin{remark}
\label{rem:up-teeth-count}
The number of up teeth in $\mathsf{M}(T)$ is at least three.
\end{remark}

\begin{definition}[Bites]
\label{def:bite}
By Remark~\ref{rem:teeth-sharing} an apex is shared by at most two down
teeth.  When two down teeth share an apex, the pair is called a
\emph{bite}, and their common apex is the \emph{apex of the bite}.  A down
tooth that belongs to a bite is a \emph{bite tooth}.  We further require the
two annular edges carrying the teeth of a bite to be \emph{non-incident}:
they share no annular vertex of $A(T)$.  Equivalently, the two bite teeth
meet only at their common apex.
\end{definition}

\begin{remark}
\label{rem:bite-face-count}
Let $B(T)$ be the subgraph of $\mathsf{M}(T)$ consisting of $A(T)$
together with all bite apexes (equivalently, $A(T)$ together with all
bite teeth), drawn as a plane graph.  For every interior face of $B(T)$
that is not a bite tooth, the number of down teeth whose apex lies in
the interior of that face is either $0$ or at least $3$.  Such an apex is
necessarily that of a singleton down tooth: every bite apex is a vertex
of $B(T)$, so it lies on a face boundary rather than in any face
interior.
\end{remark}

\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=2.3,
    ann/.style={circle, fill=black, inner sep=1.1pt},
    upv/.style={circle, draw=blue!70!black, fill=blue!15, inner sep=1.5pt},
    downv/.style={circle, draw=red!70!black, fill=red!15, inner sep=1.5pt},
    bitev/.style={circle, draw=red!70!black, fill=red!35, inner sep=1.9pt},
    cyc/.style={black, line width=1.0pt},
    tth/.style={black!55, line width=0.45pt},
    lbl/.style={font=\scriptsize},
    lead/.style={black!55, line width=0.3pt}]

% annular medial vertices forming the cycle A(T)
\foreach \k/\a in {0/105,1/75,2/45,3/15,4/-15,5/-45,6/-75,7/-105,8/-135,9/-165,10/165,11/135}
  \coordinate (v\k) at (\a:1);

% down-tooth apexes (inner region)
\coordinate (d1) at (60:0.60);
\coordinate (d2) at (30:0.60);
\coordinate (d3) at (0:0.60);
\coordinate (d4) at (-30:0.60);

% shared apex of the bite
\coordinate (p) at (0,0);

% up-tooth apexes (outer region)
\coordinate (u1) at (-60:1.35);
\coordinate (u2) at (-120:1.35);
\coordinate (u3) at (-150:1.35);
\coordinate (u4) at (180:1.35);
\coordinate (u5) at (150:1.35);
\coordinate (u6) at (120:1.35);

% annular medial cycle
\draw[cyc] (v0)--(v1)--(v2)--(v3)--(v4)--(v5)--(v6)--(v7)--(v8)--(v9)--(v10)--(v11)--cycle;

% down teeth into the right inner region
\draw[tth] (d1)--(v1) (d1)--(v2);
\draw[tth] (d2)--(v2) (d2)--(v3);
\draw[tth] (d3)--(v3) (d3)--(v4);
\draw[tth] (d4)--(v4) (d4)--(v5);

% up teeth
\draw[tth] (u1)--(v5) (u1)--(v6);
\draw[tth] (u2)--(v7) (u2)--(v8);
\draw[tth] (u3)--(v8) (u3)--(v9);
\draw[tth] (u4)--(v9) (u4)--(v10);
\draw[tth] (u5)--(v10) (u5)--(v11);
\draw[tth] (u6)--(v11) (u6)--(v0);

% the bite: two down teeth sharing the apex p
\draw[tth] (p)--(v0) (p)--(v1) (p)--(v6) (p)--(v7);

% vertices
\foreach \k in {0,...,11} \node[ann] at (v\k) {};
\foreach \u in {u1,u2,u3,u4,u5,u6} \node[upv] at (\u) {};
\foreach \dd in {d1,d2,d3,d4} \node[downv] at (\dd) {};
\node[bitev] at (p) {};

% annotations
\node[lbl, anchor=west] at (42:1.78) (Ldt) {down tooth};
\draw[lead] (Ldt.west) -- (30:0.74);

\node[lbl, anchor=east] at (150:1.86) (Lut) {up tooth};
\draw[lead] (Lut.east) -- (150:1.22);

\node[lbl] at (90:1.62) (Lbite) {bite};
\draw[lead] (Lbite.south) -- (0,0.08);

\node[lbl, anchor=west] at (-12:1.78) (L4) {region with 4 down teeth};
\draw[lead] (L4.west) -- (-9:0.80);

\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
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.}
\label{fig:medial-teeth-example}
\end{figure}

\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.82,
    ann/.style={circle, fill=black, inner sep=1.3pt},
    outv/.style={circle, draw=blue!70!black, fill=blue!12, inner sep=1.7pt},
    inv/.style={circle, draw=red!70!black, fill=red!12, inner sep=1.7pt},
    mededge/.style={black!62, line width=0.45pt},
    attach/.style={black!45, line width=0.38pt}]

\newcommand{\hexannulus}[1]{
  \coordinate (#1a0) at (90:1);
  \coordinate (#1a1) at (30:1);
  \coordinate (#1a2) at (-30:1);
  \coordinate (#1a3) at (-90:1);
  \coordinate (#1a4) at (-150:1);
  \coordinate (#1a5) at (150:1);
  \draw[mededge] (#1a0)--(#1a1)--(#1a2)--(#1a3)--(#1a4)--(#1a5)--cycle;
  \foreach \i in {0,...,5} \node[ann] at (#1a\i) {};
}

\begin{scope}[shift={(-4.8,0)}]
  \hexannulus{x}
  \coordinate (xb0) at (60:0.48);
  \coordinate (xb1) at (0:0.48);
  \coordinate (xb2) at (-60:0.48);
  \coordinate (xb3) at (-120:1.46);
  \coordinate (xb4) at (180:1.46);
  \coordinate (xb5) at (120:1.46);
  \foreach \i/\j in {0/1,1/2,2/3,3/4,4/5} {
    \draw[attach] (xb\i)--(xa\i);
    \draw[attach] (xb\i)--(xa\j);
  }
  \draw[attach] (xb5)--(xa5);
  \draw[attach] (xb5)--(xa0);
  \foreach \i in {0,1,2} \node[inv] at (xb\i) {\scriptsize I};
  \foreach \i in {3,4,5} \node[outv] at (xb\i) {\scriptsize O};
  \node at (0,-1.86) {\scriptsize \texttt{IIIOOO}};
  \node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$};
\end{scope}

\begin{scope}
  \hexannulus{y}
  \coordinate (yb0) at (60:0.48);
  \coordinate (yb1) at (0:0.48);
  \coordinate (yb2) at (-60:1.46);
  \coordinate (yb3) at (-120:0.48);
  \coordinate (yb4) at (180:1.46);
  \coordinate (yb5) at (120:1.46);
  \foreach \i/\j in {0/1,1/2,2/3,3/4,4/5} {
    \draw[attach] (yb\i)--(ya\i);
    \draw[attach] (yb\i)--(ya\j);
  }
  \draw[attach] (yb5)--(ya5);
  \draw[attach] (yb5)--(ya0);
  \foreach \i in {0,1,3} \node[inv] at (yb\i) {\scriptsize I};
  \foreach \i in {2,4,5} \node[outv] at (yb\i) {\scriptsize O};
  \node at (0,-1.86) {\scriptsize \texttt{IIOIOO}};
  \node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$};
\end{scope}

\begin{scope}[shift={(4.8,0)}]
  \hexannulus{z}
  \coordinate (zb0) at (60:0.48);
  \coordinate (zb1) at (0:0.48);
  \coordinate (zb2) at (-60:1.46);
  \coordinate (zb3) at (-120:1.46);
  \coordinate (zb4) at (180:0.48);
  \coordinate (zb5) at (120:1.46);
  \foreach \i/\j in {0/1,1/2,2/3,3/4,4/5} {
    \draw[attach] (zb\i)--(za\i);
    \draw[attach] (zb\i)--(za\j);
  }
  \draw[attach] (zb5)--(za5);
  \draw[attach] (zb5)--(za0);
  \foreach \i in {0,1,4} \node[inv] at (zb\i) {\scriptsize I};
  \foreach \i in {2,3,5} \node[outv] at (zb\i) {\scriptsize O};
  \node at (0,-1.86) {\scriptsize \texttt{IIOOIO}};
  \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
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.}
\label{fig:medial-restriction-worst-cases}
\end{figure}

\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.82,
    mededge/.style={black!38, line width=0.42pt},
    cycleedge/.style={black, line width=1.25pt},
    czero/.style={circle, draw=blue!65!black, fill=blue!18, inner sep=1.6pt},
    cone/.style={circle, draw=red!65!black, fill=red!18, inner sep=1.6pt},
    ctwo/.style={circle, draw=green!45!black, fill=green!20, inner sep=1.6pt}]

\coordinate (n12) at (90:1.45);
\coordinate (n13) at (30:1.45);
\coordinate (n14) at (-30:1.45);
\coordinate (n15) at (-90:1.45);
\coordinate (n16) at (-150:1.45);
\coordinate (n17) at (150:1.45);
\coordinate (n23) at (1.85,1.55);
\coordinate (n34) at (2.45,0.00);
\coordinate (n45) at (1.85,-1.55);
\coordinate (n56) at (-0.15,-2.45);
\coordinate (n67) at (-1.85,-1.55);
\coordinate (n27) at (-1.85,1.55);
\coordinate (n26) at (-0.30,2.55);
\coordinate (n36) at (0.30,0.55);
\coordinate (n35) at (0.45,-0.85);

\draw[mededge] (n12)--(n13);
\draw[mededge] (n12)--(n17);
\draw[mededge] (n12)--(n23);
\draw[mededge] (n12)--(n27);
\draw[mededge] (n13)--(n14);
\draw[mededge] (n13)--(n23);
\draw[mededge] (n13)--(n34);
\draw[mededge] (n14)--(n15);
\draw[mededge] (n14)--(n34);
\draw[mededge] (n14)--(n45);
\draw[mededge] (n15)--(n16);
\draw[mededge] (n15)--(n45);
\draw[mededge] (n15)--(n56);
\draw[mededge] (n16)--(n17);
\draw[mededge] (n16)--(n56);
\draw[mededge] (n16)--(n67);
\draw[mededge] (n17)--(n27);
\draw[mededge] (n17)--(n67);
\draw[mededge] (n23)--(n26);
\draw[mededge] (n23)--(n36);
\draw[mededge] (n26)--(n27);
\draw[mededge] (n26)--(n36);
\draw[mededge] (n26)--(n67);
\draw[mededge] (n27)--(n67);
\draw[mededge] (n34)--(n35);
\draw[mededge] (n34)--(n45);
\draw[mededge] (n35)--(n36);
\draw[mededge] (n35)--(n45);
\draw[mededge] (n35)--(n56);
\draw[mededge] (n36)--(n56);

\draw[cycleedge] (n12)--(n13)--(n14)--(n15)--(n16)--(n17)--cycle;

\node[czero] (v12) at (n12) {\scriptsize $12$};
\node[cone]  (v13) at (n13) {\scriptsize $13$};
\node[ctwo]  (v14) at (n14) {\scriptsize $14$};
\node[czero] (v15) at (n15) {\scriptsize $15$};
\node[ctwo]  (v16) at (n16) {\scriptsize $16$};
\node[cone]  (v17) at (n17) {\scriptsize $17$};
\node[ctwo]  (v23) at (n23) {\scriptsize $23$};
\node[cone]  (v26) at (n26) {\scriptsize $26$};
\node[ctwo]  (v27) at (n27) {\scriptsize $27$};
\node[czero] (v34) at (n34) {\scriptsize $34$};
\node[ctwo]  (v35) at (n35) {\scriptsize $35$};
\node[czero] (v36) at (n36) {\scriptsize $36$};
\node[cone]  (v45) at (n45) {\scriptsize $45$};
\node[cone]  (v56) at (n56) {\scriptsize $56$};
\node[czero] (v67) at (n67) {\scriptsize $67$};

\node[anchor=west] at (3.0,1.45) {\scriptsize colour $0$};
\node[czero] at (2.82,1.45) {};
\node[anchor=west] at (3.0,1.05) {\scriptsize colour $1$};
\node[cone] at (2.82,1.05) {};
\node[anchor=west] at (3.0,0.65) {\scriptsize colour $2$};
\node[ctwo] at (2.82,0.65) {};
\node[anchor=west, text width=2.35cm] at (2.82,-0.15)
  {\scriptsize thick cycle: annular medial cycle for source $1$};
\end{tikzpicture}
\caption{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.}
\label{fig:medial-annular-cycle-counterexample}
\end{figure}

\begin{definition}[Boundary medial vertices]
\label{def:boundary-medial-vertices}
Let $T$ be a tire tread and let $\Gamma_T$ be the corresponding dual
tire subgraph in $G^*$.  A vertex $m_e \in V(\mathsf{M}(T))$ is an
\emph{outer boundary medial vertex} if the corresponding dual edge
$e^* \in E(G^*)$ lies on the outer boundary of $\Gamma_T$.  It is an
\emph{inner boundary medial vertex} if $e^*$ lies on the inner boundary
of $\Gamma_T$.  We write
\[
    \partial_{\mathrm{out}}\mathsf{M}(T)
    \quad\text{and}\quad
    \partial_{\mathrm{in}}\mathsf{M}(T)
\]
for the two boundary sets.
\end{definition}

\begin{definition}[Medial tire restriction relation]
\label{def:medial-restriction-relation}
Let $\mathrm{Col}_3(X)$ denote the set of proper vertex
$3$-colourings of the induced subgraph on a vertex set $X$.  The
\emph{medial tire restriction relation} of $T$ is
\[
    R_T \subseteq
    \mathrm{Col}_3(\partial_{\mathrm{out}}\mathsf{M}(T))
    \times
    \mathrm{Col}_3(\partial_{\mathrm{in}}\mathsf{M}(T)),
\]
where $(\alpha,\beta) \in R_T$ exactly when $\alpha \cup \beta$
extends to a proper vertex $3$-colouring of $\mathsf{M}(T)$.
\end{definition}

\begin{remark}
The definition deliberately records boundary colourings on medial
vertices corresponding to boundary edges in the dual tire graph.  Under
Corollary~\ref{cor:tait-medial}, these are precisely edge-colouring
states on the boundary edges through which a dual tire piece meets its
parent and children.
\end{remark}

\section{Decomposition}

\begin{corollary}[Medial tire decomposition]
\label{cor:medial-tire-decomposition}
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.
\end{corollary}

\begin{proof}
By the tire-tread partition theorem of
\cite{bauerfeld-nested-tire-decompositions}, the bounded triangular
faces of $G$ are partitioned into nested tire treads, with intersections
between parent and child treads occurring only along their level-cycle
interface data.  Every edge of $G$ that is incident to a bounded face
therefore belongs to the closure of at least one tire tread, and an
edge lying in two closures lies on the interface between adjacent
treads in the tire tree.  Passing to $M(G)$ sends edges of $G$ to
medial vertices.  Thus each tread determines the induced subgraph
$\mathsf{M}(T)$ on its incident edge set, and overlaps between two such
subgraphs are exactly the medial vertices corresponding to interface
edges, namely the appropriate boundary medial vertex sets.
\end{proof}

\begin{definition}[Compatible family of medial tire colourings]
\label{def:compatible-family}
A \emph{compatible family of medial tire colourings} on
$\mathcal{T}(G,S)$ is a choice, for each tread $T$, of a proper
vertex $3$-colouring $\varphi_T$ of $\mathsf{M}(T)$ such that whenever
$T'$ is a child tread of $T$, the two colourings agree on
$
    V(\mathsf{M}(T)) \cap V(\mathsf{M}(T')).
$
\end{definition}

\begin{proposition}[Gluing criterion]
\label{prop:gluing-criterion}
The full medial graph $M(G)$ has a proper vertex $3$-colouring if and
only if the tire tree $\mathcal{T}(G,S)$ admits a compatible family of
medial tire colourings.
\end{proposition}

\begin{proof}
A proper vertex $3$-colouring of $M(G)$ restricts to a proper vertex
$3$-colouring of every induced subgraph $\mathsf{M}(T)$, and these
restrictions agree on overlaps.

Conversely, suppose a compatible family is given.  Define a colour on
each vertex $m_e$ of $M(G)$ by choosing any tread $T$ with
$m_e \in V(\mathsf{M}(T))$ and setting
$\varphi(m_e)=\varphi_T(m_e)$.  Compatibility makes this independent of
the choice of $T$.  Every medial edge of $M(G)$ is drawn in a corner of
some bounded triangular face of $G$ or along the outer boundary
interface.  The relevant incident primal edges lie together in the
closure of a single tire tread or in a shared boundary interface, where
properness is already enforced by one of the local colourings.  Hence
$\varphi$ is a proper vertex $3$-colouring of $M(G)$.
\end{proof}

\section{A medial pigeonhole programme}

The restriction relation $R_T$ records exactly the local information
needed to pass a medial $3$-colouring through a tire.  In a nested
chain
\[
    T_0 \supset T_1 \supset \cdots \supset T_k,
\]
the outer boundary state of $T_{i+1}$ must match an inner boundary
state allowed by $R_{T_i}$.  Thus a proof of the Four Colour Theorem in
this framework would follow from a structural reason that these
restriction sets cannot remain mutually disjoint along every branch of
the tire tree.

\begin{definition}[Medial boundary state]
\label{def:medial-boundary-state}
A \emph{medial boundary state} on a boundary set
$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the
subgraph induced by that boundary set, considered up to permutation of
the three colours and the dihedral symmetries of the boundary walk
when that boundary is a cycle.
\end{definition}

\begin{conjecture}[Medial chain-pigeonhole principle]
\label{conj:medial-chain-pigeonhole}
There is a function $N(k)$ such that the following holds.  Let
$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of
tire treads whose relevant boundary medial walks have length at most
$k$.  Then two adjacent restriction relations in the chain have
compatible medial boundary states after colour permutation and boundary
symmetry.  Equivalently, the chain contains a local gluing step that
cannot be obstructed by disjoint proper vertex $3$-colouring
restrictions.
\end{conjecture}

\begin{conjecture}[Medial tire route to the Four Colour Theorem]
\label{conj:medial-route-fct}
For every plane triangulation $G$ and every level source $S$, the
restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a
compatible selection of boundary states across the tire tree.  Hence
$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly
$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable.
\end{conjecture}

\begin{remark}
Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to
the Four Colour Theorem when combined with Tait's correspondence.  The
point of the formulation is not to weaken the target theorem, but to
move the obstruction into finite boundary-state restrictions carried by
annular medial tire pieces.
\end{remark}

\subsection{Kempe-cycle conservation across medial tires}

We now record an additional structure carried by proper
$3$-colourings of medial graphs.  This structure will be useful for
describing how colourings glue across level cycles.

Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph.
Let
\[
  \varphi:V(M)\to\{1,2,3\}
\]
be a proper $3$-colouring of $M$.  For a two-element colour set
$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$
induced by the vertices of colours $a$ and $b$.

Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
$M_P$ has degree $2$ in $M_P$.  Hence every component of $M_P$ is a
cycle.  We call these components the $P$-Kempe cycles of $\varphi$.

\begin{lemma}[Kempe chains are cycles]
\label{lem:kempe-cycles}
Let $G$ be a plane triangulation, let $M=M(G)$, and let
$\varphi$ be a proper $3$-colouring of $M$.  For each
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
\end{lemma}

\begin{proof}
Let $v\in V(M_P)$.  In the medial graph $M$, the vertex $v$ has degree
$4$.  Since $\varphi$ is a proper $3$-colouring, none of the neighbours
of $v$ has colour $\varphi(v)$.  Thus all four neighbours of $v$ have
one of the two colours different from $\varphi(v)$.

In the medial graph of a plane triangulation, the neighbours of a
medial vertex occur in two opposite pairs corresponding to the two
faces incident with the corresponding edge of $G$.  Around each such
triangular face, the three medial vertices receive all three colours.
Consequently, at $v$ there are exactly two neighbours of each colour
different from $\varphi(v)$.  It follows that, in the subgraph induced
by any two colours $P$, every vertex has degree $2$.  Hence each
component of $M_P$ is a cycle.
\end{proof}

Let $T$ be a medial tire region.  We regard $T$ as an annular transition
region whose boundary consists of one outer level cycle and finitely
many inner level cycles:
\[
  \partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
Here $C_0$ is the outer level cycle of $T$, and the cycles
$C_1,\ldots,C_m$ are the inner level cycles.  Each inner level cycle
$C_i$ is also the outer level cycle of the corresponding child region
in the tire tree.

The following lemma is the basic conservation principle.

\begin{lemma}[Kempe-cycle conservation across level cycles]
\label{lem:kempe-conservation}
Let $C$ be a level cycle of $M$ separating a parent side from a child
side.  Let $K$ be a $P$-Kempe cycle for some
$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$.  Then $K$ cannot enter the child side
of $C$ without also leaving it.

Equivalently, the incidences of $K$ with $C$ are paired by the
components of $K$ lying on the child side of $C$, and also paired by the
components of $K$ lying on the parent side of $C$.
\end{lemma}

\begin{proof}
By the preceding lemma, $K$ is a cycle.  The level cycle $C$ separates
the sphere into two closed regions, which we call the parent side and
the child side.  Consider the intersection of $K$ with one of these
regions.  Since $K$ is a cycle, no component of this intersection can
have exactly one boundary endpoint on $C$.  Each component is either
closed within the region, or is a path with two boundary endpoints on
$C$.  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
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
$\mathsf{M}(T)$.  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
$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe
chains} of $\varphi$.  Every vertex of colour $a$ or $b$ lies on exactly
one $P$-Kempe chain.

A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior
face of $B(T)$ that is not a tooth---namely the root face or a bite
inner-gap face of Remark~\ref{rem:bite-face-count}.  The \emph{tooth
apexes incident to} a valid face $F$ are:
\begin{itemize}
  \item the up-tooth apexes (Definition~\ref{def:annular-teeth}), when
  $F$ is the outer face;
  \item the singleton down-tooth apexes whose annular edge lies on $F$,
  when $F$ is interior---the apex on annular edge $m$ being incident to
  the innermost bite $(i,j)$ with $i<m<j$, or to the root face if there
  is none.
\end{itemize}
Bite apexes are never incident to a valid face in this sense.

For a colour pair $P=\{a,b\}$ write $\nu_P(F)$ for the number of tooth
apexes incident to $F$ that are coloured $a$ or $b$---equivalently, that
lie on a $P$-Kempe chain.  The colouring $\varphi$ is
\emph{Kempe-balanced} if $\nu_P(F)$ is even for every valid face $F$ and
every colour pair $P$.
\end{definition}

\begin{remark}[Necessity of Kempe-balance]
\label{rem:kempe-balance-necessary}
A proper $3$-colouring of $\mathsf{M}(T)$ can be part of a proper
$3$-colouring of the whole medial graph $M(G)$ only when it is
Kempe-balanced: if $\varphi$ is the restriction to $\mathsf{M}(T)$ of a
proper $3$-colouring of $M(G)$, then $\varphi$ is Kempe-balanced.
Equivalently, a colouring of $\mathsf{M}(T)$ that fails the parity
condition at some valid face and colour pair cannot extend to a proper
$3$-colouring of $M(G)$.  This is an instance of Kempe-cycle
conservation (Lemma~\ref{lem:kempe-conservation}).  The tooth apexes
incident to a valid face are boundary medial vertices
(Definition~\ref{def:boundary-medial-vertices}) lying on a single level
cycle of the tire decomposition: the up-tooth apexes lie on the outer
level cycle, and the singleton down-tooth apexes incident to an interior
non-tooth face lie on the inner level cycle bounding that face.  In the
$4$-regular graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes
up into a $P$-Kempe cycle, which by Lemma~\ref{lem:kempe-conservation}
meets each level cycle in an even number of $P$-coloured incidences; for
a given valid face these incidences are exactly its incident tooth
apexes coloured $a$ or $b$, whence $\nu_P(F)$ is even.

This argument is verified computationally.  For bite-free pieces---capped
triangulated annuli on annular cycles of length $6,8,10,12$---every proper
$3$-colouring of $M(G)$ restricts to a Kempe-balanced colouring.  The same
holds for pieces carrying a bite, including the case where singleton down
teeth lie in the bite's inner-gap face: there the inner level cycle splits
into a child level cycle per gap, and conservation across each child cycle
supplies the parity (in the checked example the three singleton down apexes
of a bite gap are a rainbow in every restriction).
\end{remark}

More generally, let $T$ be a medial tire region with boundary
\[
  \partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
For a $P$-Kempe cycle $K$, every component of $K\cap T$ is either a
cycle contained in $T$, or a path with two endpoints on
$\partial T$.  Thus the $P$-Kempe arcs inside $T$ define a pairing of
the $P$-coloured boundary incidences of
\[
  C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
This motivates the following refinement of boundary states.

\begin{definition}[Kempe-enhanced boundary state]
Let $T$ be a medial tire region with outer level cycle $C_0$ and inner
level cycles $C_1,\ldots,C_m$.  Let
\[
  \mathcal C(T)=C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
\]
A \emph{Kempe-enhanced boundary state} on $T$ consists of the following
data:
\begin{enumerate}
  \item a boundary colouring
  \[
    \alpha:V(\mathcal C(T))\to\{1,2,3\};
  \]
  \item for each colour pair
  \[
    P\in\{\{1,2\},\{2,3\},\{3,1\}\},
  \]
  a pairing $\pi_P$ of the $P$-coloured boundary incidences of
  $\mathcal C(T)$ induced by the $P$-Kempe arcs lying inside $T$.
\end{enumerate}
We write such a state as
\[
  \kappa=(\alpha,\pi_{12},\pi_{23},\pi_{31}).
\]
\end{definition}

Given a proper $3$-colouring $\varphi$ of the medial tire graph
$M(T)$, the restriction of $\varphi$ to the boundary level cycles gives
the boundary colouring $\alpha$, while the two-colour Kempe arcs inside
$T$ give the pairings $\pi_{12},\pi_{23},\pi_{31}$.  Thus $\varphi$
determines a Kempe-enhanced boundary state, denoted
\[
  \kappa_T(\varphi).
\]

\begin{definition}[Kempe-enhanced restriction relation]
The \emph{Kempe-enhanced restriction relation} of $T$ is
\[
  \mathcal K_T
  =
  \left\{
    \kappa_T(\varphi):
    \varphi \text{ is a proper }3\text{-colouring of } M(T)
  \right\}.
\]
This refines the ordinary boundary-colouring relation by recording not
only which boundary colourings extend across $T$, but also how the
two-colour Kempe cycles are routed through the annular tire region.
\end{definition}

The annular structure of a tire is useful in two distinct ways.  First,
it gives a bounded transition region between level cycles: the colouring
of the annular medial cycle controls, and in many cases determines, the
colouring of the remaining medial tire vertices.  Thus the number of
possible transition states is bounded in terms of the annular structure,
rather than the total size of the subtree below the tire.  Second, it
describes how the outer level cycle and the inner level cycles are
related by Kempe arcs.  The level cycles are the gluing interfaces, while
the annular tire is the transition operator between them.

\begin{definition}[Kempe-compatible gluing]
Let $T$ be a medial tire region and let $U$ be a child region glued to
$T$ along a common level cycle $C$.  Thus $C$ is an inner level cycle of
$T$ and the outer level cycle of $U$.

Let
\[
  \kappa_T=(\alpha_T,\pi^T_{12},\pi^T_{23},\pi^T_{31})
  \in \mathcal K_T
\]
and
\[
  \kappa_U=(\alpha_U,\pi^U_{12},\pi^U_{23},\pi^U_{31})
  \in \mathcal K_U.
\]
We say that $\kappa_T$ and $\kappa_U$ are \emph{Kempe-compatible along
$C$} if:
\begin{enumerate}
  \item the boundary colourings agree on $C$:
  \[
    \alpha_T|_{V(C)}=\alpha_U|_{V(C)};
  \]
  \item for each colour pair
  \[
    P\in\{\{1,2\},\{2,3\},\{3,1\}\},
  \]
  the pairings $\pi^T_P$ and $\pi^U_P$ compose along the
  $P$-coloured incidences of $C$ without producing an unpaired endpoint.
\end{enumerate}
When these conditions hold, the composed pairings determine a
Kempe-enhanced boundary state on the exposed boundary of
$T\cup_C U$.
\end{definition}

In these terms, gluing local colourings is not merely a matter of
matching boundary colours.  The colourings must also route their
two-colour Kempe arcs compatibly across every shared level cycle.  The
ordinary restriction relation records whether a boundary colouring can
be extended locally; the Kempe-enhanced relation additionally records
the conservation of Kempe-cycle flow through the annular transition
region.

For a tire with one outer level cycle and several inner level cycles,
\[
  \partial T=C_0\sqcup C_1\sqcup\cdots\sqcup C_m,
\]
the parent tire may correlate the boundary states on the different
inner cycles.  The Kempe-enhanced relation records this correlation as
a system of pairings among the $P$-coloured incidences of all boundary
level cycles simultaneously.  Thus one should view a medial tire as a
multi-output transition operator
\[
  \mathcal K_T:
  C_0 \leadsto (C_1,\ldots,C_m),
\]
rather than as an independent collection of binary transitions.

The guiding principle is therefore:

\begin{quote}
Level cycles are the interfaces used for gluing, while annular tire
regions are the bounded transition regions that route Kempe cycles
between those interfaces.
\end{quote}


\begin{thebibliography}{9}

\bibitem{bauerfeld-nested-tire-decompositions}
E.~Bauerfeld,
\emph{Nested Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.

\bibitem{tait-original}
P.~G. Tait,
\emph{Remarks on the colourings of maps},
Proceedings of the Royal Society of Edinburgh \textbf{10} (1880),
729--729.

\end{thebibliography}

\end{document}