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>
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@@ -53,9 +53,12 @@ isomorphic to the medial graph of the planar dual $G^*$, and proper
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$3$-vertex-colourings of $M(G)$ are equivalent to proper
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$3$-edge-colourings of the cubic dual. Thus Tait's reformulation of
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the Four Colour Theorem may be studied through proper vertex
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$3$-colourings of medial subgraphs. We define medial tire pieces,
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their boundary-state restriction relations, and a chain-pigeonhole
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conjecture for compatible medial boundary states across the tire tree.
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$3$-colourings of medial subgraphs. We define medial tire pieces and
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their boundary-state restriction relations, and show that a proper
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vertex $3$-colouring of $M(G)$ amounts to a compatible selection of
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these boundary states across the tire tree. The resulting
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pigeonhole programme for the Four Colour Theorem is developed
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in~\cite{bauerfeld-medial-pigeonhole}.
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\end{abstract}
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\maketitle
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@@ -656,340 +659,6 @@ properness is already enforced by one of the local colourings. Hence
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$\varphi$ is a proper vertex $3$-colouring of $M(G)$.
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\end{proof}
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\section{A medial pigeonhole programme}
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The restriction relation $R_T$ records exactly the local information
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needed to pass a medial $3$-colouring through a tire. In a nested
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chain
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\[
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T_0 \supset T_1 \supset \cdots \supset T_k,
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\]
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the outer boundary state of $T_{i+1}$ must match an inner boundary
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state allowed by $R_{T_i}$. Thus a proof of the Four Colour Theorem in
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this framework would follow from a structural reason that these
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restriction sets cannot remain mutually disjoint along every branch of
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the tire tree.
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\begin{definition}[Medial boundary state]
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\label{def:medial-boundary-state}
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A \emph{medial boundary state} on a boundary set
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$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the
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subgraph induced by that boundary set, considered up to permutation of
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the three colours and the dihedral symmetries of the boundary walk
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when that boundary is a cycle.
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\end{definition}
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\begin{conjecture}[Medial chain-pigeonhole principle]
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\label{conj:medial-chain-pigeonhole}
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There is a function $N(k)$ such that the following holds. Let
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$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of
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tire treads whose relevant boundary medial walks have length at most
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$k$. Then two adjacent restriction relations in the chain have
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compatible medial boundary states after colour permutation and boundary
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symmetry. Equivalently, the chain contains a local gluing step that
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cannot be obstructed by disjoint proper vertex $3$-colouring
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restrictions.
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\end{conjecture}
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\begin{conjecture}[Medial tire route to the Four Colour Theorem]
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\label{conj:medial-route-fct}
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For every plane triangulation $G$ and every level source $S$, the
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restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a
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compatible selection of boundary states across the tire tree. Hence
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$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly
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$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable.
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\end{conjecture}
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\begin{remark}
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Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to
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the Four Colour Theorem when combined with Tait's correspondence. The
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point of the formulation is not to weaken the target theorem, but to
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move the obstruction into finite boundary-state restrictions carried by
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annular medial tire pieces.
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\end{remark}
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\subsection{Kempe-cycle conservation across medial tires}
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We now record an additional structure carried by proper
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$3$-colourings of medial graphs. This structure will be useful for
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describing how colourings glue across level cycles.
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Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph.
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Let
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\[
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\varphi:V(M)\to\{1,2,3\}
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\]
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be a proper $3$-colouring of $M$. For a two-element colour set
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$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$
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induced by the vertices of colours $a$ and $b$.
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Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
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$M_P$ has degree $2$ in $M_P$. Hence every component of $M_P$ is a
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cycle. We call these components the $P$-Kempe cycles of $\varphi$.
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\begin{lemma}[Kempe chains are cycles]
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\label{lem:kempe-cycles}
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Let $G$ be a plane triangulation, let $M=M(G)$, and let
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$\varphi$ be a proper $3$-colouring of $M$. For each
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$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
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\end{lemma}
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\begin{proof}
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Let $v\in V(M_P)$. In the medial graph $M$, the vertex $v$ has degree
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$4$. Since $\varphi$ is a proper $3$-colouring, none of the neighbours
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of $v$ has colour $\varphi(v)$. Thus all four neighbours of $v$ have
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one of the two colours different from $\varphi(v)$.
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In the medial graph of a plane triangulation, the neighbours of a
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medial vertex occur in two opposite pairs corresponding to the two
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faces incident with the corresponding edge of $G$. Around each such
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triangular face, the three medial vertices receive all three colours.
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Consequently, at $v$ there are exactly two neighbours of each colour
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different from $\varphi(v)$. It follows that, in the subgraph induced
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by any two colours $P$, every vertex has degree $2$. Hence each
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component of $M_P$ is a cycle.
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\end{proof}
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Let $T$ be a medial tire region. We regard $T$ as an annular transition
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region whose boundary consists of one outer level cycle and finitely
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many inner level cycles:
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\[
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\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
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\]
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Here $C_0$ is the outer level cycle of $T$, and the cycles
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$C_1,\ldots,C_m$ are the inner level cycles. Each inner level cycle
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$C_i$ is also the outer level cycle of the corresponding child region
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in the tire tree.
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The following lemma is the basic conservation principle.
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\begin{lemma}[Kempe-cycle conservation across level cycles]
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\label{lem:kempe-conservation}
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Let $C$ be a level cycle of $M$ separating a parent side from a child
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side. Let $K$ be a $P$-Kempe cycle for some
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$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$. Then $K$ cannot enter the child side
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of $C$ without also leaving it.
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Equivalently, the incidences of $K$ with $C$ are paired by the
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components of $K$ lying on the child side of $C$, and also paired by the
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components of $K$ lying on the parent side of $C$.
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\end{lemma}
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\begin{proof}
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By the preceding lemma, $K$ is a cycle. The level cycle $C$ separates
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the sphere into two closed regions, which we call the parent side and
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the child side. Consider the intersection of $K$ with one of these
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regions. Since $K$ is a cycle, no component of this intersection can
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have exactly one boundary endpoint on $C$. Each component is either
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closed within the region, or is a path with two boundary endpoints on
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$C$. Thus every entrance through $C$ is paired with an exit through
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$C$.
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\end{proof}
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We now use these Kempe cycles to single out the colourings of a full
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medial tire graph that respect the annular tooth structure.
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\begin{definition}[Kempe-balanced colouring]
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\label{def:kempe-balanced}
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Let $\varphi$ be a proper $3$-colouring of the full medial tire graph
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$\mathsf{M}(T)$. For a colour pair $P=\{a,b\}$, let $\mathsf{M}(T)_P$ be
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the subgraph induced by the vertices of colours $a$ and $b$. Since
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$\mathsf{M}(T)$ need not be $4$-regular, the components of
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$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe
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chains} of $\varphi$. Every vertex of colour $a$ or $b$ lies on exactly
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one $P$-Kempe chain.
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A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior
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face of $B(T)$ that is not a tooth---namely the root face or a bite
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inner-gap face of Remark~\ref{rem:bite-face-count}. The \emph{tooth
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apexes incident to} a valid face $F$ are:
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\begin{itemize}
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\item the up-tooth apexes (Definition~\ref{def:annular-teeth}), when
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$F$ is the outer face;
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\item the singleton down-tooth apexes whose annular edge lies on $F$,
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when $F$ is interior---the apex on annular edge $m$ being incident to
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the innermost bite $(i,j)$ with $i<m<j$, or to the root face if there
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is none.
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\end{itemize}
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Bite apexes are never incident to a valid face in this sense.
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For a colour pair $P=\{a,b\}$ write $\nu_P(F)$ for the number of tooth
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apexes incident to $F$ that are coloured $a$ or $b$---equivalently, that
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lie on a $P$-Kempe chain. The colouring $\varphi$ is
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\emph{Kempe-balanced} if $\nu_P(F)$ is even for every valid face $F$ and
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every colour pair $P$.
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\end{definition}
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\begin{remark}[Necessity of Kempe-balance]
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\label{rem:kempe-balance-necessary}
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A proper $3$-colouring of $\mathsf{M}(T)$ can be part of a proper
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$3$-colouring of the whole medial graph $M(G)$ only when it is
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Kempe-balanced: if $\varphi$ is the restriction to $\mathsf{M}(T)$ of a
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proper $3$-colouring of $M(G)$, then $\varphi$ is Kempe-balanced.
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Equivalently, a colouring of $\mathsf{M}(T)$ that fails the parity
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condition at some valid face and colour pair cannot extend to a proper
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$3$-colouring of $M(G)$. This is an instance of Kempe-cycle
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conservation (Lemma~\ref{lem:kempe-conservation}). The tooth apexes
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incident to a valid face are boundary medial vertices
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(Definition~\ref{def:boundary-medial-vertices}) lying on a single level
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cycle of the tire decomposition: the up-tooth apexes lie on the outer
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level cycle, and the singleton down-tooth apexes incident to an interior
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non-tooth face lie on the inner level cycle bounding that face. In the
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$4$-regular graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes
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up into a $P$-Kempe cycle, which by Lemma~\ref{lem:kempe-conservation}
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meets each level cycle in an even number of $P$-coloured incidences; for
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a given valid face these incidences are exactly its incident tooth
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apexes coloured $a$ or $b$, whence $\nu_P(F)$ is even.
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This argument is verified computationally. For bite-free pieces---capped
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triangulated annuli on annular cycles of length $6,8,10,12$---every proper
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$3$-colouring of $M(G)$ restricts to a Kempe-balanced colouring. The same
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holds for pieces carrying a bite, including the case where singleton down
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teeth lie in the bite's inner-gap face: there the inner level cycle splits
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into a child level cycle per gap, and conservation across each child cycle
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supplies the parity (in the checked example the three singleton down apexes
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of a bite gap are a rainbow in every restriction).
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\end{remark}
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More generally, let $T$ be a medial tire region with boundary
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\[
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\partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
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\]
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For a $P$-Kempe cycle $K$, every component of $K\cap T$ is either a
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cycle contained in $T$, or a path with two endpoints on
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$\partial T$. Thus the $P$-Kempe arcs inside $T$ define a pairing of
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the $P$-coloured boundary incidences of
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\[
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C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
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\]
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This motivates the following refinement of boundary states.
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\begin{definition}[Kempe-enhanced boundary state]
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Let $T$ be a medial tire region with outer level cycle $C_0$ and inner
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level cycles $C_1,\ldots,C_m$. Let
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\[
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\mathcal C(T)=C_0\sqcup C_1\sqcup\cdots\sqcup C_m.
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\]
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A \emph{Kempe-enhanced boundary state} on $T$ consists of the following
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data:
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\begin{enumerate}
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\item a boundary colouring
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\[
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\alpha:V(\mathcal C(T))\to\{1,2,3\};
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\]
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\item for each colour pair
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\[
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P\in\{\{1,2\},\{2,3\},\{3,1\}\},
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\]
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a pairing $\pi_P$ of the $P$-coloured boundary incidences of
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$\mathcal C(T)$ induced by the $P$-Kempe arcs lying inside $T$.
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\end{enumerate}
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We write such a state as
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\[
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\kappa=(\alpha,\pi_{12},\pi_{23},\pi_{31}).
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\]
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\end{definition}
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Given a proper $3$-colouring $\varphi$ of the medial tire graph
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$M(T)$, the restriction of $\varphi$ to the boundary level cycles gives
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the boundary colouring $\alpha$, while the two-colour Kempe arcs inside
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$T$ give the pairings $\pi_{12},\pi_{23},\pi_{31}$. Thus $\varphi$
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determines a Kempe-enhanced boundary state, denoted
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\[
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\kappa_T(\varphi).
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\]
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\begin{definition}[Kempe-enhanced restriction relation]
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The \emph{Kempe-enhanced restriction relation} of $T$ is
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\[
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\mathcal K_T
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=
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\left\{
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\kappa_T(\varphi):
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\varphi \text{ is a proper }3\text{-colouring of } M(T)
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\right\}.
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\]
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This refines the ordinary boundary-colouring relation by recording not
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only which boundary colourings extend across $T$, but also how the
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two-colour Kempe cycles are routed through the annular tire region.
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\end{definition}
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The annular structure of a tire is useful in two distinct ways. First,
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it gives a bounded transition region between level cycles: the colouring
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of the annular medial cycle controls, and in many cases determines, the
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colouring of the remaining medial tire vertices. Thus the number of
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possible transition states is bounded in terms of the annular structure,
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rather than the total size of the subtree below the tire. Second, it
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describes how the outer level cycle and the inner level cycles are
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related by Kempe arcs. The level cycles are the gluing interfaces, while
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the annular tire is the transition operator between them.
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\begin{definition}[Kempe-compatible gluing]
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Let $T$ be a medial tire region and let $U$ be a child region glued to
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$T$ along a common level cycle $C$. Thus $C$ is an inner level cycle of
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$T$ and the outer level cycle of $U$.
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Let
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\[
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\kappa_T=(\alpha_T,\pi^T_{12},\pi^T_{23},\pi^T_{31})
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\in \mathcal K_T
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\]
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and
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\[
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\kappa_U=(\alpha_U,\pi^U_{12},\pi^U_{23},\pi^U_{31})
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\in \mathcal K_U.
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\]
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We say that $\kappa_T$ and $\kappa_U$ are \emph{Kempe-compatible along
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$C$} if:
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\begin{enumerate}
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\item the boundary colourings agree on $C$:
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\[
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\alpha_T|_{V(C)}=\alpha_U|_{V(C)};
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\]
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\item for each colour pair
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\[
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P\in\{\{1,2\},\{2,3\},\{3,1\}\},
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\]
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the pairings $\pi^T_P$ and $\pi^U_P$ compose along the
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$P$-coloured incidences of $C$ without producing an unpaired endpoint.
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\end{enumerate}
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When these conditions hold, the composed pairings determine a
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Kempe-enhanced boundary state on the exposed boundary of
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$T\cup_C U$.
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\end{definition}
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In these terms, gluing local colourings is not merely a matter of
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matching boundary colours. The colourings must also route their
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two-colour Kempe arcs compatibly across every shared level cycle. The
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ordinary restriction relation records whether a boundary colouring can
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be extended locally; the Kempe-enhanced relation additionally records
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the conservation of Kempe-cycle flow through the annular transition
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region.
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For a tire with one outer level cycle and several inner level cycles,
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\[
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\partial T=C_0\sqcup C_1\sqcup\cdots\sqcup C_m,
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\]
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the parent tire may correlate the boundary states on the different
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inner cycles. The Kempe-enhanced relation records this correlation as
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a system of pairings among the $P$-coloured incidences of all boundary
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level cycles simultaneously. Thus one should view a medial tire as a
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multi-output transition operator
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\[
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\mathcal K_T:
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C_0 \leadsto (C_1,\ldots,C_m),
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\]
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rather than as an independent collection of binary transitions.
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The guiding principle is therefore:
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\begin{quote}
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Level cycles are the interfaces used for gluing, while annular tire
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regions are the bounded transition regions that route Kempe cycles
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between those interfaces.
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\end{quote}
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\begin{thebibliography}{9}
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\bibitem{bauerfeld-nested-tire-decompositions}
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@@ -997,6 +666,11 @@ E.~Bauerfeld,
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\emph{Nested Tire Decompositions of Plane Triangulations},
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manuscript (math-research repository), 2026.
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\bibitem{bauerfeld-medial-pigeonhole}
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E.~Bauerfeld,
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\emph{The Medial Pigeonhole Programme},
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manuscript (math-research repository), 2026.
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\bibitem{tait-original}
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P.~G. Tait,
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\emph{Remarks on the colourings of maps},
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