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coloring_nested_tire_graphs: add looser chain-pigeonhole conjecture to cut_depth_label

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Adds a "Looser chain pigeonhole hypothesis" section to
cut_depth_label.tex, between Cut tires and the
"Connection to chain pigeonhole" section.

The conjecture says: for every non-trivial cut tire T, the joint
projection π(T) is non-empty, S_3-closed, and contains at least
one full S_3-orbit (so |π(T)| ≥ 6).  Chain composition through
T_1 → T_2 → ... preserves S_3-symmetry, so R_i contains a full
S_3-orbit, and R_0 ∩ R_1 contains a common orbit, contradicting G'
being a counterexample.

Status:
  - Per-tire half: provable via Prop 1.13 of paper.tex (2^n + 2(-1)^n
    colorings) + S_3-equivariance.
  - Chain half: open. Requires showing per-tire S_3-orbit structure
    composes coherently through the depth chain.

Weaker than the rainbow / König-lift conjectures (no requirement on
specific face boundary structure), with stronger empirical support
across all cut tires tested.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-26 16:15:56 -04:00
parent 702cbcecf7
commit e173b1d2d4
4 changed files with 127 additions and 15 deletions
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papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux
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\@writefile{toc}{\contentsline {paragraph}{Visualization.}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Relation to the existing tire framework.}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Example on $G'_1$ (Holton-McKay \#0, $V \setminus S$ half).}{3}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{5}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Justification of the per-tire half.}{4}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{What this would (and would not) prove.}{4}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Status.}{4}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{6}{}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:cut-depth-label}{{1}{5}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$. In each panel, blue solid edges form the face boundary at depth $d$; red dashed edges are inner spokes (depth $d - 1$); green dashed edges are outer spokes (depth $d + 1$). Vertices on the face boundary are highlighted; the rest of $G'_1$ is faded.\relax }}{6}{}\protected@file@percent }
\newlabel{fig:cut-tire}{{2}{6}}
\gdef \@abspage@last{6}
\newlabel{fig:cut-depth-label}{{1}{6}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$. In each panel, blue solid edges form the face boundary at depth $d$; red dashed edges are inner spokes (depth $d - 1$); green dashed edges are outer spokes (depth $d + 1$). Vertices on the face boundary are highlighted; the rest of $G'_1$ is faded.\relax }}{7}{}\protected@file@percent }
\newlabel{fig:cut-tire}{{2}{7}}
\gdef \@abspage@last{7}
papers/coloring_nested_tire_graphs/notes/cut_depth_label.log
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\OT1/cmr/bx/n/10.95 Jus-ti-fi-ca-tion of the per-tire half.[] \OT1/cmr/m/n/10.
95 The per-tire half is es-sen-tially Propo-si-tion 1.13 of \OT1/cmtt/m/n/10.95
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\OT1/cmr/bx/n/10.95 Sta-tus.[] \OT1/cmr/m/n/10.95 Em-pir-i-cally true for ev-e
ry cut tire in the Holton-McKay #0 ex-am-ple (tested in \OT1/cmtt/m/n/10.95 cut
[]tire[]conjecture[]tests.tex\OT1/cmr/m/n/10.95 ).
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\OT1/cmr/m/n/10.95 The pro-ce-dure mir-rors the $4$CT cut-and-reglue scheme (\O
T1/cmtt/m/n/10.95 rainbow[]proof.tex\OT1/cmr/m/n/10.95 , \OT1/cmtt/m/n/10.95 wo
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papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex
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tire in this chain.
\end{itemize}
\section*{A looser chain pigeonhole hypothesis}
The rainbow conjecture and K\"onig-lift conjecture (from
\texttt{rainbow\_proof.tex} and
\texttt{worst\_case\_proof\_sketch.tex}) are statements about
specific structured face boundaries (antipodal-chord SP). Generic
cut tires in our example do not have those specific structures, so
those conjectures don't apply directly. Empirically
(\texttt{cut\_tire\_conjecture\_tests.tex}), cut-tire projections
are nonetheless large and well-behaved:
\begin{itemize}
\item Every cut tire's joint projection $\pi(T) :=
\{(\sigma_{\mathrm{out}}, \sigma_{\mathrm{in}}) : \chi
\text{ proper}\}$ is $S_3$-closed (universal, follows from
color symmetry of proper edge $3$-colouring).
\item All $S_3$-orbits in $\pi(T)$ have size exactly $3$ (the
constant-colour orbit, when present) or $6$ (the generic
orbit using all $3$ colours); no size-$2$ stabilizer orbits
occur.
\item Non-trivial cut tires have $|\pi(T)| \gg 6$: e.g.\ at
depth~$2$ the projection has size $126 = 21 \cdot 6$
($21$ full $S_3$-orbits), and at depth~$6$ size $93 = 1
\cdot 3 + 15 \cdot 6$.
\end{itemize}
This motivates a much weaker conjecture than the rainbow/K\"onig
ones --- one that does not require special face-boundary structure
and may be easier to prove:
\begin{quote}
\textbf{Loose chain pigeonhole conjecture (cut tires).} Let
$T = T_d^{(i, f)}$ be a non-trivial cut tire (one with at least one
in or out spoke). Then the joint projection $\pi(T)$ is non-empty,
$S_3$-closed, and contains at least one full $S_3$-orbit (i.e.\
$|\pi(T)| \geq 6$, with equality only in the degenerate case where
$\pi(T)$ is a single non-constant $S_3$-orbit).
Consequently, for the chain $T_1^{(i)} \to T_2^{(i)} \to \cdots$ on
each side of a cut, the realisable cut-configuration set
$\mathcal{R}_i$ contains a full $S_3$-orbit, and the intersection
$\mathcal{R}_0 \cap \mathcal{R}_1$ contains a common $S_3$-orbit
(of size $6$), forcing $\mathcal{R}_0 \cap \mathcal{R}_1 \ne
\emptyset$.
\end{quote}
\paragraph{Justification of the per-tire half.} The per-tire half
is essentially Proposition~1.13 of \texttt{paper.tex}: a spoke-only
cut tire (face boundary a simple cycle of length $n$) has $2^n +
2(-1)^n$ proper edge $3$-colourings, which is $\ge 6$ whenever
$n \ge 3$. By $S_3$-closure, the orbit count is at least $1$,
hence $\ge 6$ distinct projections (modulo the size-$3$ constant
orbit edge case). For cut tires with non-simple face boundary
(e.g.\ $\theta(1, p, q)$-shape) similar lower bounds follow from
the chromatic polynomial of the corresponding $D(T)$ structure (see
the menagerie note for explicit closed forms when chord matchings
are present).
\paragraph{Justification of the chain half.} Chain composition
preserves $S_3$-symmetry (each step is $S_3$-equivariant), so
$\mathcal{R}_i$ inherits $S_3$-closure from the per-tire projection.
The non-trivial claim is that the orbit structure \emph{survives}
composition: an $S_3$-orbit in $\pi(T_d)$ projects to some
non-empty $S_3$-closed subset on the out-spoke side, and this
propagates layer-by-layer.
\paragraph{What this would (and would not) prove.} If the loose
conjecture holds, chain pigeonhole at the cut gives $\mathcal{R}_0
\cap \mathcal{R}_1 \neq \emptyset$, which (under the minimum-
counterexample reduction) contradicts $G'$ being a 4CT
counterexample. This is the same final conclusion as the
rainbow/K\"onig route, with weaker hypotheses but stronger
empirical support.
It would \emph{not} prove the rainbow conjecture itself (which is a
strict structural claim about the antipodal-chord SP case), nor
the K\"onig-lift conjecture's specific construction; it would only
recover the bottom-line chain pigeonhole result.
\paragraph{Status.} Empirically true for every cut tire in the
Holton-McKay \#0 example (tested in
\texttt{cut\_tire\_conjecture\_tests.tex}). No general proof.
The per-tire half is essentially provable via Prop~1.13 plus
$S_3$-equivariance. The chain half is the genuinely open question
--- it requires showing that the per-tire $S_3$-orbit structure
composes coherently through the depth chain.
\section*{Connection to chain pigeonhole / 4CT reducibility}
The procedure mirrors the $4$CT cut-and-reglue scheme