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>
This commit is contained in:
@@ -0,0 +1,442 @@
|
||||
%% filename: amsart-template.tex
|
||||
%% American Mathematical Society
|
||||
%% AMS-LaTeX v.2 template for use with amsart
|
||||
%% ====================================================================
|
||||
|
||||
\documentclass{amsart}
|
||||
|
||||
\usepackage{amssymb}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{backgrounds}
|
||||
|
||||
\newtheorem{theorem}{Theorem}[section]
|
||||
\newtheorem{lemma}[theorem]{Lemma}
|
||||
\newtheorem{corollary}[theorem]{Corollary}
|
||||
\newtheorem{proposition}[theorem]{Proposition}
|
||||
\newtheorem{conjecture}[theorem]{Conjecture}
|
||||
|
||||
\theoremstyle{definition}
|
||||
\newtheorem{definition}[theorem]{Definition}
|
||||
\newtheorem{example}[theorem]{Example}
|
||||
\newtheorem{xca}[theorem]{Exercise}
|
||||
|
||||
\theoremstyle{remark}
|
||||
\newtheorem{remark}[theorem]{Remark}
|
||||
|
||||
\numberwithin{equation}{section}
|
||||
|
||||
\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}
|
||||
Reference in New Issue
Block a user