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Account for the outer face in the Heawood face-sum identity

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The bounded-face sum omits the outer face at outer-boundary vertices, so
restrict the gluing identity to interior vertices (where all cluster
interfaces live) and recover a colouring by carrying a single +/-1 label
on the unbounded face f_inf, giving Heawood's identity on the full cubic
dual for the Tait step.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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didericis
2026-06-17 01:33:07 -04:00
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4 changed files with 56 additions and 42 deletions
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papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux
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\newlabel{def:boundary-sequences}{{3.3}{3}} \newlabel{def:boundary-sequences}{{3.3}{3}}
\newlabel{def:heawood-compatible}{{3.4}{3}} \newlabel{def:heawood-compatible}{{3.4}{3}}
\citation{Heawood1898}
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\newlabel{eq:heawood-face-sum-dual}{{3.1}{4}} \newlabel{eq:heawood-face-sum-dual}{{3.1}{4}}
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\newlabel{prop:two-sided-decomposition}{{3.6}{4}} \newlabel{prop:two-sided-decomposition}{{3.6}{4}}
\bibcite{Heawood1898}{1} \bibcite{Heawood1898}{1}
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\bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tires}{3} \bibcite{bauerfeld-nested-tires}{3}
\bibcite{bauerfeld-medial-tires}{4} \bibcite{bauerfeld-medial-tires}{4}
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papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log
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papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf
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papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex
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@@ -287,20 +287,33 @@ rotational senses in which $T$ and $T'$ traverse $\gamma$.
\begin{remark} \begin{remark}
\label{rem:compat-is-heawood} \label{rem:compat-is-heawood}
Compatibility along $\gamma$ at $v$ is exactly the statement that the Call $v$ \emph{interior} if it is not incident to the outer face of
full incident-face sum at $v$ --- over the parent's annular faces $\Pi_G$. For an interior vertex every incident face is bounded, and
together with the child's --- vanishes mod $3$: compatibility along $\gamma$ at $v$ is exactly the statement that the
incident-face sum at $v$ --- over the parent's annular faces together
with the child's --- vanishes mod $3$:
\begin{equation} \begin{equation}
\label{eq:heawood-face-sum-dual} \label{eq:heawood-face-sum-dual}
\sum_{f \ni v} \lambda(f) \;\equiv\; 0 \pmod 3 \sum_{f \ni v} \lambda(f) \;\equiv\; 0 \pmod 3
\qquad\text{for every vertex } v \in V(G). \qquad\text{for every interior vertex } v \in V(G),
\end{equation} \end{equation}
Since $\gamma$ carries all faces of $G$ incident to $v$ between the two the sum ranging over the bounded faces incident to $v$. The interfaces
tires, a family of Heawood face-labellings that is pairwise compatible of $\mathcal{T}(G, S)$ are interior level cycles, so cluster
along every interface of $\mathcal{T}(G, S)$ assembles into a single compatibility only ever constrains interior vertices and is untouched by
$\{+1,-1\}$ face-labelling of $G$ satisfying the outer face.
\eqref{eq:heawood-face-sum-dual} at every vertex, hence (by Tait) a
proper $4$-vertex-colouring of $G$. To pass from \eqref{eq:heawood-face-sum-dual} to a colouring one must
account for the outer face: an outer-boundary vertex is incident to the
unbounded face $f_\infty$, whose label is omitted from the bounded sum.
Extend $\lambda$ by a single label $\lambda(f_\infty) \in \{+1, -1\}$ on
$f_\infty$. Then a family of Heawood face-labellings that is pairwise
compatible along every interface of $\mathcal{T}(G, S)$ assembles into a
$\{+1,-1\}$ labelling of \emph{all} faces of $G$ for which
$\sum_{f \ni v} \lambda(f) \equiv 0 \pmod 3$ holds at every vertex ---
the outer-boundary vertices now carrying $\lambda(f_\infty)$ in their
sum. This is Heawood's face-sum identity \cite{Heawood1898} for a
proper $3$-edge-colouring of the full cubic dual of $G$, hence (by Tait)
a proper $4$-vertex-colouring of $G$.
\end{remark} \end{remark}
\subsection*{Why the programme runs between nested clusters} \subsection*{Why the programme runs between nested clusters}