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+\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{Heawood Restrictions on Nested Tire Graph Duals} + +% author one information +\author{Eric Bauerfeld} +\address{} +\curraddr{} +\email{} +\thanks{} + +\subjclass[2010]{Primary } + +\keywords{plane graph, triangulation, plane depth, level edge, dual graph, +tire graph, Heawood number} + +\date{} + +\dedicatory{} + +\begin{abstract} +%% TODO: abstract. Following \cite{bauerfeld-nested-tires}, which establishes +%% the basic vocabulary of tire graphs and dual depth, we study the Heawood +%% (mod-3 / face-sum) restrictions imposed on the duals of nested tire graphs. +\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. Thus a +minimal counterexample to the Four Colour Theorem -- a smallest triangulation +admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph +admitting no proper $3$-edge-colouring. + +This paper continues the series studying that structure through the +lens of \emph{nested level duals}. The foundational vocabulary --- +level sources, levels, the inner planar dual $G'$ and its dual depth, +and tire graphs --- is developed in the companion paper +\cite{bauerfeld-nested-tires}; we refer to that paper for those +definitions and rely on them throughout. In particular we use, +without restating, the notions of: +\begin{itemize} + \item \emph{level source} $S$ and $G$-vertex levels $\ell_G(v)$; + \item the inner planar dual $G'$ + (\cite[Definition~1.3]{bauerfeld-nested-tires}); + \item \emph{dual depth} $\delta_G(d_f)$ + (\cite[Definition~1.4]{bauerfeld-nested-tires}); + \item \emph{tire graph} $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ + with outer/inner boundaries and annular edges + (\cite[Definition~1.5]{bauerfeld-nested-tires}); + \item the \emph{tire-component lemma} + (\cite[Lemma~1.8]{bauerfeld-nested-tires}); and + \item the \emph{tire-tread partition theorem} + (\cite[Theorem~1.9]{bauerfeld-nested-tires}). +\end{itemize} + +Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) +with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ +and $G$ has $2n - 4$ triangular faces. + +%% TODO: state the Heawood restriction this paper studies. The relevant +%% classical input is Heawood's face-sum identity \cite{Heawood1898}; the goal +%% here is to record what it forces on the dual of a (nested) tire graph. + +\section{Heawood restrictions on the tire dual} +\label{sec:heawood-restrictions} + +%% TODO: main development. + +\begin{thebibliography}{9} + +\bibitem{Heawood1898} +P.~J.~Heawood, +\emph{On the four-colour map theorem}, +Quart. J.~Pure Appl. Math. \textbf{29} (1898), 270--285. + +\bibitem{bauerfeld-depth} +E.~Bauerfeld, +\emph{Plane Depth}, +manuscript (math-research repository), 2026. + +\bibitem{bauerfeld-nested-tires} +E.~Bauerfeld, +\emph{Nested Tire Decompositions of Plane Triangulations}, +manuscript (math-research repository), 2026. + +\bibitem{bauerfeld-nested-tire-duals} +E.~Bauerfeld, +\emph{Coloring Nested Tire Dual Graphs}, +manuscript (math-research repository), 2026. + +\end{thebibliography} + +\end{document}