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master ... split-medial-pigeonhole-programme

Author SHA1 Message Date
didericis d3fc4bfc4c Split medial pigeonhole programme into its own paper 
Move Section 5 of "Medial Tire Decompositions of Plane Triangulations"
into a new standalone paper, "The Medial Pigeonhole Programme", which
cites the medial tire paper for its terminology and notation. Convert
the three cross-references that pointed into earlier sections (annular
teeth, bite-face-count, boundary medial vertices) into citations.

Remove Section 5 from the medial tire paper and update its abstract to
drop the moved chain-pigeonhole claim, pointing to the follow-up paper.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
 2026-06-14 21:08:06 -04:00
10 changed files with 1656 additions and 384 deletions
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papers/medial_pigeonhole_programme/paper.aux
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\relax
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-nested-tire-decompositions}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A medial pigeonhole programme}}{1}{}\protected@file@percent }
\newlabel{def:medial-boundary-state}{{2.1}{2}}
\newlabel{conj:medial-chain-pigeonhole}{{2.2}{2}}
\newlabel{conj:medial-route-fct}{{2.3}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Kempe-cycle conservation across medial tires}}{2}{}\protected@file@percent }
\newlabel{lem:kempe-cycles}{{3.1}{2}}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\citation{bauerfeld-medial-tire}
\newlabel{lem:kempe-conservation}{{3.2}{3}}
\newlabel{def:kempe-balanced}{{3.3}{3}}
\newlabel{rem:kempe-balance-necessary}{{3.4}{3}}
\bibcite{bauerfeld-medial-tire}{1}
\bibcite{bauerfeld-nested-tire-decompositions}{2}
\bibcite{tait-original}{3}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{12.7778pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent }
\gdef \@abspage@last{6}
papers/medial_pigeonhole_programme/paper.fdb_latexmk
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%% filename: amsart-template.tex
%% American Mathematical Society
%% AMS-LaTeX v.2 template for use with amsart
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\documentclass{amsart}
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\usetikzlibrary{backgrounds}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
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\begin{document}
\title{The Medial Pigeonhole Programme}
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Kempe chain, Four Colour Theorem}
\date{}
\dedicatory{}
\begin{abstract}
Building on the medial tire decomposition of a plane triangulation, we
formulate a pigeonhole programme for the Four Colour Theorem in medial
terms. Each tire carries a boundary-state restriction relation, and a
proper vertex $3$-colouring of the full medial graph is a compatible
selection of these boundary states across the tire tree. We state a
chain-pigeonhole conjecture asserting that the restriction relations
cannot remain mutually disjoint along every branch, and we refine the
boundary states by recording how two-colour Kempe cycles are routed
through each annular tire region. This yields a Kempe-enhanced
restriction relation and a notion of Kempe-compatible gluing along level
cycles.
\end{abstract}
\maketitle
\section{Introduction}
This paper continues the medial tire programme begun
in~\cite{bauerfeld-medial-tire}. We use freely the terminology and
notation introduced there. For a plane triangulation $G$ with fixed
embedding, $M(G)$ denotes the full medial graph, and the tire-tree
decomposition $\mathcal{T}(G,S)$ at a level source $S$
of~\cite{bauerfeld-nested-tire-decompositions} induces a decomposition
of $M(G)$ into full medial tire graphs $\mathsf{M}(T)$, one for each
tread $T$, glued along their boundary medial vertex sets
$\partial_{\mathrm{out}}\mathsf{M}(T)$ and
$\partial_{\mathrm{in}}\mathsf{M}(T)$. We also use the annular medial
cycle $A(T)$, its up and down teeth and their apexes, the bites and the
auxiliary plane graph $B(T)$, and the medial tire restriction relation
$R_T$ of~\cite{bauerfeld-medial-tire}.
By the Tait--medial correspondence of~\cite{bauerfeld-medial-tire},
proper vertex $3$-colourings of $M(G)$ are in natural bijection with
proper $3$-edge-colourings of the cubic planar dual $G^*$. Thus the
Four Colour Theorem is the assertion that the full medial graph of every
plane triangulation is properly vertex $3$-colourable, and the medial
tire decomposition turns this into a question about how local boundary
colourings compose across the tire tree.
\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}
\section{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 in the sense of~\cite{bauerfeld-medial-tire}. The
\emph{tooth apexes incident to} a valid face $F$ are:
\begin{itemize}
\item the up-tooth apexes (\cite{bauerfeld-medial-tire}), 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
(\cite{bauerfeld-medial-tire}) 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-medial-tire}
E.~Bauerfeld,
\emph{Medial Tire Decompositions of Plane Triangulations},
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{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}
papers/medial_tire_decompositions_of_plane_triangulations/paper.aux
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\citation{bauerfeld-medial-pigeonhole}
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@@ -13,7 +14,7 @@
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\newlabel{rem:teeth-sharing}{{3.5}{3}}
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\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 }
@@ -26,25 +27,17 @@
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@@ -53,9 +53,12 @@ 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.
$3$-colourings of medial subgraphs. We define medial tire pieces and
their boundary-state restriction relations, and show that a proper
vertex $3$-colouring of $M(G)$ amounts to a compatible selection of
these boundary states across the tire tree. The resulting
pigeonhole programme for the Four Colour Theorem is developed
in~\cite{bauerfeld-medial-pigeonhole}.
\end{abstract}
\maketitle
@@ -656,340 +659,6 @@ 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}
@@ -997,6 +666,11 @@ E.~Bauerfeld,
\emph{Nested Tire Decompositions of Plane Triangulations},
manuscript (math-research repository), 2026.
\bibitem{bauerfeld-medial-pigeonhole}
E.~Bauerfeld,
\emph{The Medial Pigeonhole Programme},
manuscript (math-research repository), 2026.
\bibitem{tait-original}
P.~G. Tait,
\emph{Remarks on the colourings of maps},