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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.

The classical input is Heawood's face-sum identity \cite{Heawood1898}:
for any proper $3$-edge-colouring of a cubic plane graph $H$, assigning
each face of $H$ a number in $\{+1, -1\}$ can be done so that the labels
around every vertex of $H$ sum to $0 \pmod 3$.  In the triangulation
$G$ dual to $H$ this becomes a $\{+1, -1\}$ labelling of the
\emph{faces} of $G$ whose incident-face sum at every vertex of $G$
vanishes mod $3$.  Our aim is to record what this restriction forces
along the boundary cycles of a nested tire graph, and to formulate a
chain-pigeonhole programme in this Heawood labelling parallel to the
medial programme of \cite{bauerfeld-medial-tires}.

\section{Heawood restrictions on the tire dual}
\label{sec:heawood-restrictions}

We work inside a fixed nested tire decomposition $\mathcal{T}(G, S)$ of
$G$ from a single-vertex level source $S$ \cite{bauerfeld-nested-tires},
and use the tire data $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ with
annular faces $F_{\mathrm{ann}}$, outer boundary $B_{\mathrm{out}}$, and
inner boundary $B_{\mathrm{in}}$
(\cite[Definition~1.5]{bauerfeld-nested-tires}).  Since $O$ is
outerplanar, every vertex of a tire lies on $B_{\mathrm{out}}$ or on the
inner-boundary walk $B_{\mathrm{in}}$; a tire has no interior vertices.

\begin{definition}[Heawood face-labelling of a tire]
\label{def:heawood-labelling}
A \emph{Heawood face-labelling} of a tire graph $T$ is a map
\[
   \lambda : F_{\mathrm{ann}} \longrightarrow \{+1, -1\}
\]
assigning a sign to each annular face of $T$.  For a vertex
$v \in V(T)$, write $F_{\mathrm{ann}}(v) \subseteq F_{\mathrm{ann}}$ for
the set of annular faces of $T$ incident to $v$, and define the
\emph{induced vertex value}
\[
   \lambda^{\!*}(v) \;:=\; \sum_{f \in F_{\mathrm{ann}}(v)} \lambda(f)
   \;\;\bmod 3 \;\in\; \{0, 1, -1\}.
\]
The value $\lambda^{\!*}(v)$ is the \emph{partial} face-sum at $v$ taken
over the annular faces of $T$ alone, not over all faces of $G$ incident
to $v$.
\end{definition}

\begin{remark}
\label{rem:no-interior-constraint}
Because a tire has no interior vertices, every annular face of $T$ is
incident to $B_{\mathrm{out}} \cup B_{\mathrm{in}}$, and a Heawood
face-labelling is subject to \emph{no} internal constraint: all
$2^{|F_{\mathrm{ann}}|}$ sign assignments are admissible.  The Heawood
restriction is felt only on the two boundary cycles, through the induced
vertex values $\lambda^{\!*}$.
\end{remark}

\begin{definition}[Induced boundary sequences]
\label{def:boundary-sequences}
Let $\lambda$ be a Heawood face-labelling of $T$.  Reading the vertices
of $B_{\mathrm{out}}$ in clockwise order $v_0, v_1, \dots, v_{p-1}$, the
\emph{outer Heawood sequence} of $(T, \lambda)$ is
\[
   \sigma_{\mathrm{out}}(T, \lambda)
   \;:=\; \bigl(\lambda^{\!*}(v_0), \dots, \lambda^{\!*}(v_{p-1})\bigr)
   \;\in\; \{0, 1, -1\}^{p}.
\]
Reading the inner-boundary walk $B_{\mathrm{in}}$ in clockwise order
$w_0, \dots, w_{q-1}$ gives the \emph{inner Heawood sequence}
$\sigma_{\mathrm{in}}(T, \lambda) \in \{0, 1, -1\}^{q}$.  The
\emph{Heawood restriction relation} of $T$ is the set
\[
   R_T \;:=\; \bigl\{\,
     \bigl(\sigma_{\mathrm{out}}(T, \lambda),\,
            \sigma_{\mathrm{in}}(T, \lambda)\bigr)
     \;:\; \lambda : F_{\mathrm{ann}} \to \{+1, -1\}
   \,\bigr\}
\]
of all (outer, inner) sequence pairs realisable by a single
face-labelling, read up to rotation and the global sign-flip
$\lambda \mapsto -\lambda$ (equivalently
$\sigma \mapsto -\sigma$).
\end{definition}

\begin{definition}[Heawood compatibility across an interface]
\label{def:heawood-compatible}
Let $T$ be a tire and $T' \in \mathcal{T}(G, S)$ a child of $T$, so the
outer boundary cycle $B_{\mathrm{out}}^{(T')}$ coincides with a bounded
face of $O^{(T)}$; let $\gamma$ be this shared cycle, of length $L$, and
let $v$ range over its vertices.  Heawood face-labellings $\lambda$ of
$T$ and $\lambda'$ of $T'$ are \emph{compatible along $\gamma$} if at
every shared vertex $v$,
\[
   \lambda^{\!*}(v) + (\lambda')^{\!*}(v) \;\equiv\; 0 \pmod 3,
\]
i.e.\ $0$ is paired with $0$ and $+1$ with $-1$.  Equivalently, the
inner Heawood sequence of $T$ on $\gamma$ is the pointwise negation
mod $3$ of the outer Heawood sequence of $T'$ on $\gamma$, after
reversing one of the two clockwise readings to account for the opposite
rotational senses in which $T$ and $T'$ traverse $\gamma$.
\end{definition}

\begin{remark}
\label{rem:compat-is-heawood}
Compatibility along $\gamma$ at $v$ is exactly the statement that the
full incident-face sum at $v$ --- over the parent's annular faces
together with the child's --- vanishes mod $3$:
\begin{equation}
\label{eq:heawood-face-sum-dual}
   \sum_{f \ni v} \lambda(f) \;\equiv\; 0 \pmod 3
   \qquad\text{for every vertex } v \in V(G).
\end{equation}
Since $\gamma$ carries all faces of $G$ incident to $v$ between the two
tires, a family of Heawood face-labellings that is pairwise compatible
along every interface of $\mathcal{T}(G, S)$ assembles into a single
$\{+1,-1\}$ face-labelling of $G$ satisfying
\eqref{eq:heawood-face-sum-dual} at every vertex, hence (by Tait) a
proper $4$-vertex-colouring of $G$.
\end{remark}

\begin{conjecture}[Heawood chain-pigeonhole principle]
\label{conj:heawood-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 tires in $\mathcal{T}(G, S)$ whose shared interface
cycles have length at most $k$.  Then two adjacent Heawood restriction
relations $R_{T_i}, R_{T_{i+1}}$ in the chain admit compatible
face-labellings along their shared interface
(Definition~\ref{def:heawood-compatible}), after rotation and global
sign-flip.  Equivalently, the chain contains a local gluing step that
cannot be obstructed by disjoint Heawood boundary restrictions.
\end{conjecture}

\begin{conjecture}[Heawood tire route to the Four Colour Theorem]
\label{conj:heawood-route-fct}
For every plane triangulation $G$ and every level source $S$, the
Heawood restriction relations $\{R_T : T \in \mathcal{T}(G, S)\}$ admit
a selection of face-labellings that is compatible along every interface
of the tire tree.  By Remark~\ref{rem:compat-is-heawood} this yields a
$\{+1,-1\}$ face-labelling of $G$ satisfying
\eqref{eq:heawood-face-sum-dual}, hence $G$ is properly
$4$-vertex-colourable.
\end{conjecture}

%% TODO: realisability of $R_T$ per tire; counting / pigeonhole bound
%% giving $N(k)$; orientation/reversal bookkeeping on $\gamma$.

\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-medial-tires}
E.~Bauerfeld,
\emph{Medial 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}