coloring_nested_tire_graphs: counterexamples to the loose conjecture as stated
Before attempting to prove the loose chain pigeonhole conjecture
("|π(T)| ≥ 6" for every non-trivial cut tire), looked for
counterexamples and found TWO in the existing empirical data:
(d, face) = (1, 1): 1 out spoke, |π(T)| = 3, orbit size [3].
(d, face) = (4, 0): 1 out spoke, |π(T)| = 3, orbit size [3].
Reason: a cut tire with exactly k = 1 in/out spoke has σ ∈ {1,2,3}.
S_3 acts with stabilizer of size 2 on any single-color σ, so the
orbit has size 6/2 = 3, never 6. The "|π(T)| ≥ 6" claim is
automatically false for k = 1 tires.
For k ≥ 2: σ can use 1 color (size-3 orbit) or ≥ 2 colors
(size-6 orbit). |π(T)| ≥ 6 requires at least one multi-color σ to
extend, which is not automatic but typically holds.
Three refined conjectures proposed:
1. Restrict to k ≥ 2 spokes (avoids the trivial counterexample).
2. Weaken to "non-empty and S_3-closed" (very weak; needs the
chain composition to preserve non-emptiness).
3. Just describe orbit sizes 3 or 6 (no useful claim).
The two found counterexamples are at "side" faces in the chain;
they don't break the bottom-line chain pigeonhole because the main
chain runs through larger faces.
To find harder counterexamples: look for k ≥ 2 cut tires whose face
boundary forces all spoke colors equal (= |π(T)| = 3 with k ≥ 2).
Such examples might exist but weren't found in the current data.
Recommended next step: restrict the conjecture to k ≥ 2 and re-run
the empirical sweep.
Note: loose_conjecture_counterexamples.tex (3 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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\documentclass[11pt]{article}
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\usepackage{amsmath,amssymb,amsthm}
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\usepackage{graphicx}
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\usepackage{geometry}
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\usepackage{booktabs}
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\geometry{margin=1in}
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\title{Counterexamples to the loose chain pigeonhole conjecture\\
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(and what they suggest about the right statement)}
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\author{}
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\date{}
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\newtheorem*{obs}{Observation}
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\begin{document}
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\maketitle
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\section*{The conjecture as stated}
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From \texttt{cut\_depth\_label.tex}, the loose chain pigeonhole
|
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conjecture says:
|
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\begin{quote}
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For every non-trivial cut tire $T$ (with $\ge 1$ in/out spoke), the
|
||||
joint projection $\pi(T)$ is non-empty, $S_3$-closed, and contains
|
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at least one full $S_3$-orbit (i.e.\ $|\pi(T)| \ge 6$).
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\end{quote}
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\section*{Counterexamples already in the data}
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Re-examining the cut tire tests on $G'_1$ of Holton-McKay \#0
|
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(\texttt{cut\_tire\_conjecture\_tests.tex}):
|
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|
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\begin{center}
|
||||
\small
|
||||
\begin{tabular}{lr|rrr|r|cl}
|
||||
\toprule
|
||||
$d$ & face & out & in & total spokes & $|\pi(T)|$ & orbit sizes & \\
|
||||
\midrule
|
||||
$1$ & $0$ & $5$ & $0$ & $5$ & $93$ & $[3, 6^{15}]$ & ✓\\
|
||||
$1$ & $1$ & $1$ & $0$ & $1$ & $3$ & $[3]$ & \textbf{✗}\\
|
||||
$2$ & $0$ & $4$ & $3$ & $7$ & $126$ & $[6^{21}]$ & ✓\\
|
||||
$2$ & $1$ & $4$ & $3$ & $7$ & $126$ & $[6^{21}]$ & ✓\\
|
||||
$4$ & $0$ & $1$ & $0$ & $1$ & $3$ & $[3]$ & \textbf{✗}\\
|
||||
$4$ & $1$ & $2$ & $1$ & $3$ & $21$ & $[3, 6^3]$ & ✓\\
|
||||
$6$ & $0$ & $3$ & $2$ & $5$ & $93$ & $[3, 6^{15}]$ & ✓\\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
Two non-trivial cut tires have $|\pi(T)| = 3 < 6$:
|
||||
$(d, \text{face}) = (1, 1)$ and $(4, 0)$. Both have exactly $1$
|
||||
spoke. In each case, $\pi(T)$ is the $3$-element $S_3$-orbit of a
|
||||
single-color spoke configuration --- which has $S_3$-stabilizer of
|
||||
order $2$ (the two permutations fixing the chosen color and swapping
|
||||
the others), so the orbit has size $|S_3| / 2 = 3$.
|
||||
|
||||
\section*{Why this happens (and is general)}
|
||||
|
||||
For a cut tire with exactly $k$ in/out spokes, the projection
|
||||
$\pi(T) \subseteq \{1, 2, 3\}^k$. $S_3$ acts by permuting colors.
|
||||
The $S_3$-orbit of $\sigma \in \{1,2,3\}^k$ has size:
|
||||
\begin{itemize}
|
||||
\item $3$ if $\sigma$ uses exactly $1$ color (stabilizer of order $2$);
|
||||
\item $6$ if $\sigma$ uses $\geq 2$ distinct colors (trivial stabilizer).
|
||||
\end{itemize}
|
||||
|
||||
For $k = 1$: every $\sigma \in \{1, 2, 3\}$ uses exactly $1$ color,
|
||||
so every orbit has size $3$, and $|\pi(T)| \in \{0, 3\}$. The
|
||||
conjecture's ``$|\pi(T)| \ge 6$'' is automatically false for any
|
||||
non-empty $\pi(T)$ on a $1$-spoke tire.
|
||||
|
||||
For $k \ge 2$, $\sigma$ can use $1$ or $\geq 2$ colors:
|
||||
\begin{itemize}
|
||||
\item Single-color $\sigma$ contributes a size-$3$ orbit.
|
||||
\item Multi-color $\sigma$ contributes size-$6$ orbits.
|
||||
\end{itemize}
|
||||
$|\pi(T)| \ge 6$ requires at least one multi-color $\sigma$, which
|
||||
is not automatic but typically holds when $k$ is large or the tire
|
||||
allows enough flexibility.
|
||||
|
||||
\section*{Why the conjecture's chain half can still fail}
|
||||
|
||||
Even if we restrict to $k \ge 2$ cut tires (avoiding the trivial
|
||||
$k = 1$ counterexample), the chain pigeonhole conclusion is not
|
||||
immediate:
|
||||
|
||||
\begin{itemize}
|
||||
\item Chain composition involves \emph{compatibility} between
|
||||
adjacent cut tires. A full $S_3$-orbit at one layer projects
|
||||
through the in/out-spoke ↔ face-boundary bijection to some
|
||||
subset at the adjacent layer; this projection might land in
|
||||
a size-$3$ orbit even if the source was size-$6$.
|
||||
\item The chain at $d_{\max}$ terminates (no in spokes); the
|
||||
innermost cut tire's structure constrains everything outward.
|
||||
\item Sufficiency for $\mathcal{R}_0 \cap \mathcal{R}_1 \neq
|
||||
\emptyset$ requires the chain to preserve the $S_3$-orbit
|
||||
structure all the way to the cut, on both sides, and have a
|
||||
common orbit there.
|
||||
\end{itemize}
|
||||
|
||||
The counterexamples at $1$-spoke tires don't break the chain at
|
||||
those depths because they're at non-essential ``side'' faces
|
||||
(face $1$ at $d = 1$, face $0$ at $d = 4$; the main chain runs
|
||||
through the larger faces). But they show the per-tire claim
|
||||
is not universal.
|
||||
|
||||
\section*{Refined conjectures}
|
||||
|
||||
In view of these counterexamples, three natural refinements:
|
||||
|
||||
\begin{enumerate}
|
||||
\item \textbf{Restrict to $\ge 2$ spokes.} ``For every cut tire
|
||||
$T$ with $\geq 2$ in/out spokes total, $|\pi(T)| \ge 6$ and
|
||||
$\pi(T)$ contains a full $S_3$-orbit.'' This avoids the
|
||||
$k = 1$ trivial case. Still empirically open whether
|
||||
$k = 2$ cut tires always have $\ge 6$ projections.
|
||||
|
||||
\item \textbf{Weaken to non-emptiness.} ``For every non-trivial
|
||||
cut tire, $\pi(T)$ is non-empty and $S_3$-closed.'' This
|
||||
is much weaker but might still suffice for chain pigeonhole
|
||||
\emph{provided} chain composition preserves non-emptiness ---
|
||||
which is itself the hard step.
|
||||
|
||||
\item \textbf{Allow size-$3$ orbits.} ``For every non-trivial cut
|
||||
tire, $\pi(T)$ is non-empty and $S_3$-closed; the orbits in
|
||||
$\pi(T)$ have size $3$ or $6$.'' This is just a description
|
||||
of the empirical findings, not a useful conjecture --- chain
|
||||
pigeonhole at a size-$3$-orbit layer is more restrictive than
|
||||
at a size-$6$ one.
|
||||
\end{enumerate}
|
||||
|
||||
\section*{Where to look for harder counterexamples}
|
||||
|
||||
The two counterexamples found are at low-spoke tires. More
|
||||
substantive counterexamples would look like:
|
||||
|
||||
\begin{itemize}
|
||||
\item A $k \ge 2$ spoke cut tire whose face boundary is structured
|
||||
such that only single-color $\sigma$'s extend, leaving
|
||||
$|\pi(T)| = 3$. This would require the face boundary to
|
||||
force all spoke colors equal --- an unusual constraint.
|
||||
\item A chain in which adjacent compatibility kills all size-$6$
|
||||
orbits, leaving only size-$3$ at the cut layer. This would
|
||||
be a chain-level counterexample to the bottom-line
|
||||
pigeonhole.
|
||||
\end{itemize}
|
||||
|
||||
Neither is observed in the present data. But the data is from a
|
||||
single $6$-cut on a single Holton-McKay graph. A broader empirical
|
||||
sweep (multiple cuts, multiple Holton-McKay graphs, larger $G'$)
|
||||
might surface harder counterexamples.
|
||||
|
||||
\paragraph{Suggested next step.} Restrict the conjecture to
|
||||
$k \ge 2$ tires and re-run the empirical test. If still empirically
|
||||
true after broader sweeps, attempt to prove the restricted form via
|
||||
the existing Prop 1.13 lower bound (which gives $\ge 2^n + 2(-1)^n$
|
||||
colorings for spoke-only cut tires of cycle length $n$) plus a
|
||||
combinatorial argument that the multi-color $\sigma$'s are non-empty.
|
||||
|
||||
\end{document}
|
||||
Reference in New Issue
Block a user