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coloring_nested_tire_graphs: remove cut_tire_chain_pigeonhole note (per user)

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Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-26 15:48:40 -04:00
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papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.aux
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\relax
\@writefile{toc}{\contentsline {paragraph}{(Step 1) Reduction by minimality.}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{(Step 2) The induced cut configuration.}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{(Step 3) Gluing.}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{(Step 4) Layered description of $\mathcal {R}_i$ via cut tires.}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{(Step 5) Chain pigeonhole at the cut.}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Concrete next steps.}{4}{}\protected@file@percent }
\gdef \@abspage@last{4}
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papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex
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\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{graphicx}
\usepackage{geometry}
\usepackage{booktabs}
\geometry{margin=1in}
\title{Chain pigeonhole on cut tires:\\
a sketch and honest assessment}
\author{}
\date{}
\newtheorem*{prop}{Proposition}
\newtheorem*{obs}{Observation}
\newtheorem*{conj}{Conjecture}
\newtheorem*{lem}{Lemma}
\begin{document}
\maketitle
\section*{Setup}
Take $G'$ the cubic planar dual of a maximal planar graph $G$.
Suppose $G'$ is a minimum counterexample to the $4$-colour theorem ---
no proper $3$-edge-colouring of $G'$ exists, but every smaller cubic
planar graph admits one.
By cyclic edge-connectivity ($G$ internally $6$-connected implies $G'$
cyclically $6$-edge-connected), pick a $6$-edge cut $C \subseteq
E(G')$ partitioning $V(G')$ into $S$ and $V \setminus S$, both
non-trivial. Form $G'_0$ and $G'_1$ as in
\texttt{cut\_depth\_label.tex} by removing $C$ and attaching pendant
edges at degree-$2$ vertices.
For the cleanest setting, assume $C$ is a \emph{matching cut} (each
boundary vertex on each side has exactly $1$ cut edge, so each side
attaches $6$ pendants). Apply the BFS depth labelling: pendants get
depth $0$; each edge adjacent (sharing a vertex) to a depth-$d$ edge
gets depth $d + 1$.
For each $d > 0$, the cut tires at $(d, f)$ (Definition in
\texttt{cut\_depth\_label.tex}) layer $G'_i$ concentrically around
the cut.
\section*{The argument, step by step}
\paragraph{(Step 1) Reduction by minimality.}
Each $G'_i$ has $|S_i| + 6 < |V(G')|$ vertices (assuming
$|V \setminus S_i| > 6$, which holds in the matching-cut case for any
non-degenerate cut). By minimality of $G'$, each $G'_i$ is properly
$3$-edge-colourable. Let $\chi_i : E(G'_i) \to \{1, 2, 3\}$ be any
such colouring.
\paragraph{(Step 2) The induced cut configuration.}
The $6$ depth-$0$ pendant edges of $G'_i$ correspond bijectively to
the $6$ cut edges of $C$ (each pendant replaces a cut edge). Let
$\sigma_i := \chi_i|_{\text{depth-}0\text{ edges of } G'_i} \in
\{1, 2, 3\}^6$, indexed by the cut edges. This is the
\emph{boundary configuration} at the cut.
\paragraph{(Step 3) Gluing.}
A proper $3$-edge-colouring of $G'$ exists iff there exists a single
colouring of $C$ that extends to both sides, i.e.\ iff some
$\sigma \in \{1, 2, 3\}^6$ is achievable as both $\sigma_0$ for some
$\chi_0$ and $\sigma_1$ for some $\chi_1$. Let
\[
\mathcal{R}_i := \{\sigma_i : \chi_i \text{ a proper edge $3$-colouring of } G'_i\}.
\]
Then $G'$ has a proper $3$-edge-colouring iff
$\mathcal{R}_0 \cap \mathcal{R}_1 \ne \emptyset$. Assuming $G'$ is a
counterexample, this intersection is empty.
\paragraph{(Step 4) Layered description of $\mathcal{R}_i$ via cut tires.}
Each cut tire $T_d^{(i, f)}$ has its own ``ring projection''
constraints. Define:
\begin{itemize}
\item $\pi_{\mathrm{in}}(T_d^{(i, f)})$: the projection of $\chi_i$
onto the depth-$(d-1)$ inner spokes of the cut tire. For
$d = 1$, this is exactly $\sigma_i$ restricted to those
pendants whose boundary vertex sits on $f$'s boundary.
\item $\pi_{\mathrm{out}}(T_d^{(i, f)})$: projection onto
depth-$(d+1)$ outer spokes.
\end{itemize}
Adjacent cut tires share layers: outer spokes of $T_d$ are inner
spokes of $T_{d+1}$ (when their faces are appropriately adjacent in
the embedding). So the chain of cut tires at depths $1, 2, \dots,
d_{\max}$ has consistency constraints
$\pi_{\mathrm{out}}(T_d) = \pi_{\mathrm{in}}(T_{d+1})$
along each chain.
\paragraph{(Step 5) Chain pigeonhole at the cut.}
$\mathcal{R}_i$ is determined by the chain of cut tires on side $i$:
a $\sigma_i$ is achievable iff there exists a consistent sequence
of cut-tire colourings from the deep interior outward to the cut
that projects to $\sigma_i$ on the depth-$0$ pendants.
Chain pigeonhole says: if at each step $d$, the cut tire's
achievable inner-projection set $\pi_{\mathrm{in}}(T_d)$ is
sufficiently large (containing enough $S_3$-orbits), then
$\mathcal{R}_0 \cap \mathcal{R}_1 \ne \emptyset$, contradicting that
$G'$ is a counterexample.
\section*{What this needs to be a proof}
The argument above sketches the \emph{shape} of a proof. The
non-trivial parts are:
\subsection*{(a) Chain consistency: well-definedness of the chain}
For the chain $T_1 \to T_2 \to \dots \to T_{d_{\max}}$ to be
well-defined, each $T_d$ must have $\ge 1$ face, and adjacent tires
must share layers cleanly. Obstacles:
\begin{itemize}
\item The depth-$d$ subgraph $H_d$ may have no faces (if $H_d$ is a
tree or empty). The empirical example
(\texttt{cut\_depth\_label.tex}) shows $H_d$ has $1$--$3$
faces at each depth $1 \le d \le 7$, but this is not
guaranteed in general.
\item Multiple faces at the same depth mean the chain forks; chain
pigeonhole becomes a tree-pigeonhole.
\item Face boundary walks need not be simple cycles --- they can
revisit vertices, as in the existing tire definition's
treatment of cut-vertices.
\end{itemize}
\subsection*{(b) Quantitative chain pigeonhole}
Even with a clean chain, we need a quantitative argument that
$|\pi_{\mathrm{in}}(T_d)|$ is large enough at each step to force
non-empty intersection with the adjacent tire. This is the
\emph{same} chain pigeonhole question studied in
\texttt{rainbow\_proof.tex} and
\texttt{worst\_case\_proof\_sketch.tex}, now applied to the
cut-tire chain.
The two open conjectures that would close this step are:
\begin{itemize}
\item \textbf{Rainbow conjecture
(\texttt{rainbow\_proof.tex}, Conj 1.5):} for the antipodal-chord
SP case, the inner-spoke projection support equals the
perms-per-half set $\mathcal{P}_m$. For cut tires this would
mean each cut tire's $\pi_{\mathrm{in}}$ saturates a known
$S_3$-symmetric set.
\item \textbf{König-lift conjecture
(\texttt{worst\_case\_proof\_sketch.tex}, Conj
\emph{t2-induces-partition}):} adjacent tires induce
$\gamma$-face partitions whose K\"onig lifts give a Latin
intersection of size $\ge 6$. The
face-pair-connection refinement
(\texttt{k9\_surviving\_partitions.tex}) corrects the naive
candidate partition.
\end{itemize}
Neither is fully proved. See
\texttt{two\_approaches\_comparison.tex} for the comparison.
\subsection*{(c) Cut-tire-specific issues}
Cut tires differ from the tires of \texttt{paper.tex} in important
ways:
\begin{itemize}
\item Cut tires are derived from the \emph{depth labelling} on
$G'_i$, not from a primal level structure on $G$. The
correspondence to primal tires (Defs 1.15--1.17) is by
analogy, not by direct identification.
\item The depth-$d$ subgraph $H_d$ is generally \emph{not} cubic
and may not even be connected. Its faces may behave
differently from $G'$'s faces.
\item The cut tire's $T'_{\mathrm{ann}}$-analogue (= face boundary
of $f$ in $H_d$) is a closed walk in $H_d$, not a cycle of
$G'$. Its structure depends on the depth labelling.
\end{itemize}
So even with the rainbow/K\"onig conjectures proved for primal tires,
their transfer to cut tires requires verification.
\section*{Empirical check: chain length and tire structure}
For the example tire chain on $G'_1$ of Holton-McKay \#0
($d = 1, \ldots, 7$):
\begin{center}
\small
\begin{tabular}{c|cccc}
\toprule
$d$ & \# faces in $H_d$ & largest face length & inner spokes & outer spokes \\
\midrule
$1$ & $2$ & $12$ & $5$ & $4$ \\
$2$ & $2$ & $7$ & $4$ & $3$ \\
$3$ & $3$ & $2$ & $2$ & $2$ \\
$4$ & $2$ & $8$ & $2$ & $5$ \\
$5$ & $2$ & $14$ & $4$ & $6$ \\
$6$ & $1$ & $12$ & $7$ & $1$ \\
\bottomrule
\end{tabular}
\end{center}
\noindent
Observations:
\begin{itemize}
\item Chain length $\le 7$ in this example.
\item Face counts and sizes vary irregularly --- depth $3$ has
three small ($2$-edge) faces; depth $5$ has a face of length
$14$.
\item Total inner $+$ outer spokes at each depth ranges from
$4$ to $10$.
\end{itemize}
This irregularity is the structural obstacle to clean chain
pigeonhole: a uniform bound on $|\pi_{\mathrm{in}}|$ across depths
seems unlikely. The actual chain pigeonhole would need to handle
depth-by-depth structure.
\section*{Net assessment}
The chain pigeonhole argument on cut tires is \emph{structurally
sound but technically open}. It:
\begin{itemize}
\item Gives a clean reformulation of the 4CT reducibility
question in terms of cut-derived layered structure.
\item Maps directly onto the existing
\texttt{paper.tex} tire framework via the
depth-as-distance-to-cut analogy.
\item Inherits all the open conjectures from
\texttt{rainbow\_proof.tex} and
\texttt{worst\_case\_proof\_sketch.tex} (chain pigeonhole at
each layer, intersection non-emptiness, etc.).
\item Adds new technical issues specific to the cut-tire setting
(depth subgraphs $H_d$ may degenerate, irregular face
structure across depths).
\end{itemize}
\paragraph{Concrete next steps.}
\begin{enumerate}
\item Verify that the cut-tire chain is well-defined on the
$6$ Holton-McKay graphs and a few other test cases (each
$G'_i$ has cut tires at every depth $d$ in some range; no
empty $H_d$).
\item Compute $\pi_{\mathrm{in}}(T_d^{(i)})$ for each cut tire and
check the rainbow $S_3$-orbit appears at the cut layer ($d = 1$).
\item Check pairwise compatibility at the cut between $G'_0$ and
$G'_1$: do $\mathcal{R}_0$ and $\mathcal{R}_1$ overlap? If
empirically yes for all $6$ Holton-McKay graphs, that's
evidence; if no for some, that's a falsification of the
chain argument as currently stated.
\end{enumerate}
Step 3 is the cleanest empirical test --- it's just an extension of
the step-$2$ pairwise compatibility analysis
(\texttt{tire\_fiber\_step2.tex}) to the cut-tire / 4CT setting.
\end{document}
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