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Run Heawood pigeonhole between nested connected tire clusters

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Add the two-sided cluster decomposition proposition: a vertex's full
Heawood face-sum splits as exactly one child-cluster contribution plus
one parent-cluster contribution (the at-most-two-clusters bound makes the
pairing binary and complete). Explain why this fails per-tire -- a vertex
on many same-depth tires has only a fragment of its face-star in any one
tire -- and recast the chain-pigeonhole and 4CT conjectures to nested
clusters with a cluster restriction relation.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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didericis
2026-06-17 01:10:30 -04:00
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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:heawood-compatible}{{3.4}{3}}
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\bibcite{Heawood1898}{1}
\bibcite{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tires}{3}
\bibcite{bauerfeld-medial-tires}{4}
\bibcite{bauerfeld-nested-tire-duals}{5}
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\newlabel{eq:heawood-face-sum-dual}{{3.1}{4}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
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\newlabel{rem:why-clusters}{{3.7}{5}}
\newlabel{conj:heawood-chain-pigeonhole}{{3.8}{5}}
\newlabel{conj:heawood-route-fct}{{3.9}{5}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent }
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@@ -303,34 +303,130 @@ $\{+1,-1\}$ face-labelling of $G$ satisfying
proper $4$-vertex-colouring of $G$.
\end{remark}
\subsection*{Why the programme runs between nested clusters}
The vanishing condition \eqref{eq:heawood-face-sum-dual} at a vertex $v$
is a constraint on the \emph{full} face-star of $v$. To run a
pigeonhole between two objects --- a child and a parent --- we need that
full sum to split as exactly two one-sided contributions, so that each
vertex label is the combination of a single child value and a single
parent value. This is true at the level of connected tire clusters, and
\emph{false} at the level of individual tires. Extend a Heawood
face-labelling to a connected tire cluster $\mathsf{K}$ by labelling
every annular face of every tire of $\mathsf{K}$, and for $v \in
V(\mathsf{K})$ write
\[
\lambda^{\!*}_{\mathsf{K}}(v) \;:=\;
\sum_{f} \lambda(f) \;\bmod 3,
\]
the sum over the annular faces of $\mathsf{K}$ incident to $v$.
\begin{proposition}[Two-sided cluster decomposition at a vertex]
\label{prop:two-sided-decomposition}
Let $v \in V(G)$ have level $\ell = \ell_G(v)$, and let
$\mathsf{K}_{\ell}$ and $\mathsf{K}_{\ell-1}$ be the at most two
connected tire clusters containing $v$, of depths $\ell$ and $\ell-1$
respectively (Proposition~\ref{prop:two-clusters-per-vertex}). Then the
bounded faces of $G$ incident to $v$ partition into the annular faces of
$\mathsf{K}_{\ell}$ at $v$ and the annular faces of $\mathsf{K}_{\ell-1}$
at $v$, and
\[
\sum_{f \ni v} \lambda(f)
\;\equiv\;
\lambda^{\!*}_{\mathsf{K}_{\ell}}(v) +
\lambda^{\!*}_{\mathsf{K}_{\ell-1}}(v)
\pmod 3 .
\]
Each one-sided value $\lambda^{\!*}_{\mathsf{K}_d}(v)$ is the
\emph{complete} sum over all depth-$d$ faces at $v$, so the Heawood
condition \eqref{eq:heawood-face-sum-dual} at $v$ reads
\[
\lambda^{\!*}_{\mathsf{K}_{\ell}}(v) +
\lambda^{\!*}_{\mathsf{K}_{\ell-1}}(v) \;\equiv\; 0 \pmod 3 ,
\]
a pairing between the single child cluster $\mathsf{K}_{\ell}$ and the
single parent cluster $\mathsf{K}_{\ell-1}$. (When $\ell = 0$, or when
$v$ bounds no depth-$\ell$ face, only one term is present.)
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:two-clusters-per-vertex} (Step~1) every bounded
face incident to $v$ has depth $\ell-1$ or $\ell$, partitioning the
incident faces by depth; by Step~2 all depth-$\ell$ faces at $v$ lie in
the single cluster $\mathsf{K}_{\ell}$ and all depth-$(\ell-1)$ faces at
$v$ in $\mathsf{K}_{\ell-1}$. Hence the depth-$\ell$ part is exactly the
annular faces of $\mathsf{K}_{\ell}$ at $v$, the depth-$(\ell-1)$ part
those of $\mathsf{K}_{\ell-1}$, and summing $\lambda$ over the two parts
gives the identity; \eqref{eq:heawood-face-sum-dual} is its vanishing.
\end{proof}
\begin{remark}[Failure at the tire level]
\label{rem:why-clusters}
Proposition~\ref{prop:two-sided-decomposition} is what makes the binary
parent/child pairing possible, and it requires the cluster. A vertex
$v$ may lie on many depth-$\ell$ tires --- the unbounded case of
Section~\ref{sec:tire-clusters} --- and the per-tire value
$\lambda^{\!*}(v)$ of Definition~\ref{def:heawood-labelling} then records
only the faces of \emph{one} tire at $v$, a fragment of $v$'s face-star.
No single child tire carries the complete depth-$\ell$ sum, so the label
$\sum_{f \ni v}\lambda(f)$ cannot be written as one child value plus one
parent value, and per-tire compatibility
(Definition~\ref{def:heawood-compatible}) fails to assemble to
\eqref{eq:heawood-face-sum-dual}. Clustering repairs this:
Proposition~\ref{prop:two-clusters-per-vertex} guarantees exactly one
cluster meets $v$ on each side, so $\lambda^{\!*}_{\mathsf{K}_{\ell}}(v)$
is the complete child contribution and
$\lambda^{\!*}_{\mathsf{K}_{\ell-1}}(v)$ the complete parent
contribution. Every vertex label is then realised as the combination of
a single child-cluster value with a single parent-cluster value, and the
pigeonhole programme below chains \emph{nested connected tire clusters}
rather than individual tires.
\end{remark}
We write $R_{\mathsf{K}}$ for the \emph{cluster Heawood restriction
relation}: the set of (outer, inner) boundary Heawood sequence pairs
realisable by a face-labelling of $\mathsf{K}$, defined as in
Definition~\ref{def:boundary-sequences} but with the outer and inner
boundaries of the cluster and the complete one-sided values
$\lambda^{\!*}_{\mathsf{K}}$ in place of a single tire's, read up to
rotation and global sign-flip. By
Proposition~\ref{prop:two-sided-decomposition} two nested clusters are
compatible along their shared interface exactly when the inner sequence
of the parent is the pointwise negation mod $3$ of the outer sequence of
the child (after the orientation reversal of
Definition~\ref{def:heawood-compatible}).
\begin{conjecture}[Heawood chain-pigeonhole principle]
\label{conj:heawood-chain-pigeonhole}
There is a function $N(k)$ such that the following holds. Let
\[
T_0 \supset T_1 \supset \cdots \supset T_{N(k)}
\mathsf{K}_0 \supset \mathsf{K}_1 \supset \cdots \supset
\mathsf{K}_{N(k)}
\]
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.
be a nested chain of connected tire clusters in $\mathcal{T}(G, S)$ whose
shared interfaces have length at most $k$. Then two adjacent cluster
restriction relations $R_{\mathsf{K}_i}, R_{\mathsf{K}_{i+1}}$ in the
chain admit compatible face-labellings along their shared interface,
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]
\begin{conjecture}[Heawood cluster 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.
cluster Heawood restriction relations
$\{R_{\mathsf{K}} : \mathsf{K} \text{ a connected tire cluster}\}$ admit
a selection of face-labellings that is compatible along every cluster
interface. By Proposition~\ref{prop:two-sided-decomposition} and
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 $\gamma$.
%% TODO: realisability of $R_{\mathsf{K}}$ per cluster; counting /
%% pigeonhole bound giving $N(k)$; orientation/reversal bookkeeping on
%% the shared interface.
\begin{thebibliography}{9}