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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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2026-06-17 01:33:07 -04:00
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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}
\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 $3$:
Call $v$ \emph{interior} if it is not incident to the outer face of
$\Pi_G$. For an interior vertex every incident face is bounded, and
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}
\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).
\qquad\text{for every interior 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 $G$.
the sum ranging over the bounded faces incident to $v$. The interfaces
of $\mathcal{T}(G, S)$ are interior level cycles, so cluster
compatibility only ever constrains interior vertices and is untouched by
the outer face.
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}
\subsection*{Why the programme runs between nested clusters}