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\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}
