From 7cd1f0d08d3691e1af16df1a6e71f37356112126 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 16:00:15 -0400 Subject: [PATCH] coloring_nested_tire_graphs: chain pigeonhole on pendant-redefined cut tires MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit New note cut_tire_chain_pigeonhole.tex (4 pages) examining what chain pigeonhole looks like under the redefined cut tire (face boundary + labelled pendants). KEY CHANGE: each cut tire is now structurally isomorphic to a partial tire dual D(T), so all results from paper.tex / rainbow_proof.tex / worst_case_proof_sketch.tex / k9_surviving_partitions.tex transfer directly without re-derivation. ARGUMENT SHAPE: Setup: min counterexample G', 6-edge cut, depth labelling, cut tires at each depth. Reduction: minimality ⇒ G'_i 3-edge-colorable ⇒ boundary configurations σ_0, σ_1. Layered: σ_i = π_out({T_1^{(i, f)}}). Chain compatibility: out spoke of T_d ↔ face boundary edge of T_{d-1} via the parent-graph correspondence. Pigeonhole: if at each layer the projection support contains S_3-symmetric structure, the chain propagates and forces R_0 ∩ R_1 ≠ ∅. WHAT'S NEW UNDER THE REDEFINITION: 1. Direct result transfer: cut tires LITERALLY are partial tire duals; no translation overhead. 2. Cubicity restored: face-boundary vertices have degree 3 in the cut tire (degree 2 in H_d + 1 pendant). 3. Combinatorial rigidity: cut tire data = face + degree-2 boundary vertices + in/out classification. WHAT STAYS OPEN: (a) Chain pigeonhole at each layer: same conjectures (rainbow, König-lift) gate the argument. (b) Chain well-definedness: trivial H_d faces (length 2), degree-> 2 boundary vertices not getting pendants. (c) Depth-by-depth variability: no uniform bound on |π_out| across depths. ASSESSMENT: strict improvement over the previous cut tire definition (no transfer overhead, cubicity restored), but the hard step remains the same as in the partial-tire-dual framework. Concrete next step: cut-tire analogue of tire_fiber_step2 — for each Holton-McKay graph, build cut tire chains on both sides of a 6-cut and check R_0 ∩ R_1 empirically. Co-Authored-By: Claude Opus 4.7 --- .../notes/cut_tire_chain_pigeonhole.aux | 3 + .../notes/cut_tire_chain_pigeonhole.log | 347 ++++++++++++++++++ .../notes/cut_tire_chain_pigeonhole.pdf | Bin 0 -> 182817 bytes .../notes/cut_tire_chain_pigeonhole.tex | 244 ++++++++++++ 4 files changed, 594 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.log create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.aux b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.aux new file mode 100644 index 0000000..c99fb68 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.aux @@ -0,0 +1,3 @@ +\relax 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LaTeX Error: Unicode character ↔ (U+2194) + not set up for use with LaTeX. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.85 Symmetrically: in spokes of $T_d$ ↔ + specific face-boundary edges of +You may provide a definition with +\DeclareUnicodeCharacter + + +Overfull \hbox (0.66916pt too wide) in paragraph at lines 95--99 +[]\OT1/cmr/bx/n/10.95 Proposition 1.13 (edge--vertex col-or-ing bi-jec-tion): \ +OT1/cmr/m/n/10.95 For spoke-only cut tires (face bound- + [] + +LaTeX Font Info: Font shape `OT1/cmtt/bx/n' in size <10.95> not available +(Font) Font shape `OT1/cmtt/m/n' tried instead on input line 103. + +Overfull \hbox (5.55577pt too wide) in paragraph at lines 109--112 +[]\OT1/cmr/bx/n/10.95 Face-pair-connection struc-tural de-scrip-tion (\OT1/cmtt +/m/n/10.95 k9[]surviving[]partitions.tex\OT1/cmr/bx/n/10.95 ): \OT1/cmr/m/n/10. +95 Cut tires + [] + +[2] [3] [4] (./cut_tire_chain_pigeonhole.aux) ) +Here is how much of TeX's memory 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The cut tire then has + fewer than $|f|$ spokes. +\end{itemize} + +\subsection*{(c) Chain length and depth-by-depth variability} + +In the example, the chain has variable face counts ($1$--$3$) and +face lengths ($2$--$14$) across depths. A uniform bound on +$|\pi_{\mathrm{out}}|$ across depths still seems unlikely; the chain +would have to handle depth-by-depth structure individually. + +\section*{Net assessment} + +The redefinition (cut tire $=$ face boundary $+$ labelled pendants +at degree-$2$ vertices) is a \textbf{strict improvement} over the +previous formulation, because: + +\begin{enumerate} + \item It identifies each cut tire with a partial tire dual + $D(T)$, so all results from \texttt{paper.tex} and downstream + notes transfer directly. + \item It restores cubicity at face-boundary vertices, which the + previous definition broke. + \item The data needed for chain compatibility (face $f$, degree-$2$ + boundary vertices, in/out classification) is finite and + combinatorial. +\end{enumerate} + +But the fundamental hard step is unchanged: chain pigeonhole at +each layer reduces to the same open conjectures (rainbow / König- +lift) that gate the partial-tire-dual chain pigeonhole. The +redefinition doesn't make those easier; it makes the transfer free +of charge. + +\paragraph{Concrete next step.} Empirically verify on the $6$ +Holton-McKay graphs: for each, pick a matching $6$-cut, build cut +tire chains on both sides, compute $\mathcal{R}_0$ and +$\mathcal{R}_1$, check whether $\mathcal{R}_0 \cap \mathcal{R}_1 +\neq \emptyset$. By 4CT (which we're \emph{not} assuming for the +proof) we know it does; the question is whether the cut-tire chain +machinery successfully identifies the overlap structurally, or +whether (after composition) the chain pigeonhole inequality fails +empirically. This is the analogue of \texttt{tire\_fiber\_step2.tex} +for the cut-tire setting. + +\end{document}