Add Heawood chain-pigeonhole programme to tire-dual paper

Define a +/-1 Heawood face-labelling of a tire, its induced boundary
Heawood sequences and restriction relation, and interface compatibility
(0<->0, +1<->-1 = vertex face-sum vanishes mod 3). State the Heawood
chain-pigeonhole conjecture and a tire route to the Four Colour Theorem,
parallel to the medial programme.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux

@@ -5,17 +5,30 @@
 \citation{bauerfeld-nested-tires}
 \citation{bauerfeld-nested-tires}
 \citation{bauerfeld-nested-tires}
+\citation{Heawood1898}
+\citation{bauerfeld-medial-tires}
+\citation{bauerfeld-nested-tires}
+\citation{bauerfeld-nested-tires}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Heawood restrictions on the tire dual}}{2}{}\protected@file@percent }
+\newlabel{sec:heawood-restrictions}{{2}{2}}
+\newlabel{def:heawood-labelling}{{2.1}{2}}
+\newlabel{rem:no-interior-constraint}{{2.2}{2}}
+\newlabel{def:boundary-sequences}{{2.3}{2}}
+\newlabel{def:heawood-compatible}{{2.4}{2}}
 \bibcite{Heawood1898}{1}
 \bibcite{bauerfeld-depth}{2}
 \bibcite{bauerfeld-nested-tires}{3}
-\bibcite{bauerfeld-nested-tire-duals}{4}
+\bibcite{bauerfeld-medial-tires}{4}
+\bibcite{bauerfeld-nested-tire-duals}{5}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{12.7778pt}
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 \newlabel{tocindent2}{0pt}
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-\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
+\newlabel{rem:compat-is-heawood}{{2.5}{3}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Heawood restrictions on the tire dual}}{1}{}\protected@file@percent }
+\newlabel{eq:heawood-face-sum-dual}{{2.1}{3}}
-\newlabel{sec:heawood-restrictions}{{2}{1}}
+\newlabel{conj:heawood-chain-pigeonhole}{{2.6}{3}}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{1}{}\protected@file@percent }
+\newlabel{conj:heawood-route-fct}{{2.7}{3}}
-\gdef \@abspage@last{1}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent }
+\gdef \@abspage@last{3}

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papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex

@@ -87,14 +87,149 @@ 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
+The classical input is Heawood's face-sum identity \cite{Heawood1898}:
-%% classical input is Heawood's face-sum identity \cite{Heawood1898}; the goal
+for any proper $3$-edge-colouring of a cubic plane graph $H$, assigning
-%% here is to record what it forces on the dual of a (nested) tire graph.
+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}
 
-%% TODO: main development.
+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}
 
@@ -113,6 +248,11 @@ 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},