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 +\@writefile{toc}{\contentsline {paragraph}{Concrete next step.}{4}{}\protected@file@percent } +\gdef \@abspage@last{4} diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.log b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.log new file mode 100644 index 0000000..922733c --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.log @@ -0,0 +1,347 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 16:00 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**cut_tire_chain_pigeonhole.tex 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LaTeX Error: Unicode character ⇒ (U+21D2) + not set up for use with LaTeX. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.49 counterexample ⇒ + $\mathcal{R}_0 \cap \mathcal{R}_1 = \emptyset$. +You may provide a definition with +\DeclareUnicodeCharacter + + +Overfull \hbox (2.10077pt too wide) in paragraph at lines 79--84 +[]\OT1/cmr/m/n/10.95 This is the chain com-pat-i-bil-ity con-straint: $\OML/cmm +/m/it/10.95 T[]$\OT1/cmr/m/n/10.95 's out-spoke pro-jec-tion $\OML/cmm/m/it/10. +95 ^^Y[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 T[]\OT1/cmr/m/n/10.95 ) \OMS/cm +sy/m/n/10.95 ^^R f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 +2\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 3\OMS/cmsy/m/n/10.95 g[]$ + [] + +[1 + +{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] + +! 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 you used: + 3247 strings out of 478268 + 48400 string characters out of 5846347 + 349617 words of memory out of 5000000 + 21436 multiletter control sequences out of 15000+600000 + 477826 words of font info for 60 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 55i,5n,62p,242b,218s stack positions out of 10000i,1000n,20000p,200000b,200000s +{/usr/local/texlive/2022/texmf-d +ist/fonts/enc/dvips/cm-super/cm-super-ts1.enc} +Output written on cut_tire_chain_pigeonhole.pdf (4 pages, 182817 bytes). +PDF statistics: + 88 PDF objects out of 1000 (max. 8388607) + 53 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 1 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf new file mode 100644 index 0000000..fb88766 Binary files /dev/null and b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex new file mode 100644 index 0000000..18b17b2 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex @@ -0,0 +1,244 @@ +\documentclass[11pt]{article} +\usepackage{amsmath,amssymb,amsthm} +\usepackage{graphicx} +\usepackage{geometry} +\usepackage{booktabs} +\geometry{margin=1in} + +\title{Chain pigeonhole on cut tires (pendant-redefined version)} +\author{} +\date{} + +\newtheorem*{prop}{Proposition} +\newtheorem*{lem}{Lemma} +\newtheorem*{conj}{Conjecture} +\newtheorem*{obs}{Observation} + +\begin{document} +\maketitle + +\section*{Setup recap} + +Take $G'$ the cubic planar dual of a maximal planar graph $G$. +Suppose $G'$ is a minimum counterexample to 4CT. Pick a $6$-edge +cut $C \subseteq E(G')$ (a matching cut), form $G'_0, G'_1$ with +pendant edges as in \texttt{cut\_depth\_label.tex}, BFS-label edges +by depth from the pendants. + +The crucial change under the \emph{redefined} cut tire definition: +each cut tire $T_d^{(i, f)}$ is structurally isomorphic to a partial +tire dual $D(T)$ in \texttt{paper.tex}. Its face boundary is the +$T'_{\mathrm{ann}}$-analogue, and its labelled pendants (out spokes +at degree-$2$ vertices incident to a depth-$(d-1)$ edge; in spokes +for depth-$(d+1)$) are the analogue of $D(T)$'s leaves. + +\section*{The chain pigeonhole argument} + +\subsection*{Reduction by minimality} + +By minimality of $G'$, each $G'_i$ (after pendant addition) is +strictly smaller than $G'$ and admits a proper $3$-edge-colouring +$\chi_i : E(G'_i) \to \{1, 2, 3\}$. Restricting $\chi_i$ to the $6$ +depth-$0$ pendants gives a boundary configuration $\sigma_i \in +\{1, 2, 3\}^6$. + +A proper $3$-edge-colouring of $G'$ exists iff some $\sigma$ is +achievable as both $\sigma_0$ and $\sigma_1$, i.e.\ +$\mathcal{R}_0 \cap \mathcal{R}_1 \neq \emptyset$ where +$\mathcal{R}_i := \{\sigma_i \mid \chi_i \text{ proper}\}$. $G'$ a +counterexample ⇒ $\mathcal{R}_0 \cap \mathcal{R}_1 = \emptyset$. + +\subsection*{Layered decomposition via cut tires} + +The depth-$0$ pendants of $G'_i$ are precisely the \emph{out spokes} +of the cut tires at depth $1$ (one out spoke per depth-$0$ edge). +So $\sigma_i = \pi_{\mathrm{out}}(\{T_1^{(i, f)}\})$, where +$\pi_{\mathrm{out}}$ projects the global colouring onto all out +spokes of depth-$1$ cut tires combined. + +More generally, the cut-tire chain at depths $d = 1, 2, \ldots, +d_{\max}$ partitions the edges of $G'_i$ by depth. Each cut tire +$T_d^{(i, f)}$ has $\chi_i$-restricted colouring that: +\begin{itemize} + \item Properly colours the face boundary (a closed walk in $H_d$), + \item Colours out spokes (representing depth-$(d-1)$ edges in + $G'_i$), + \item Colours in spokes (representing depth-$(d+1)$ edges). +\end{itemize} + +\subsection*{Compatibility between adjacent tire layers} + +An out spoke of $T_d$ at boundary vertex $v$ represents a specific +depth-$(d-1)$ edge $e^* = vw$ in $G'_i$. This edge $e^*$ is a +\emph{face boundary edge} of some cut tire at depth $d - 1$ (since +$e^*$ has depth $d - 1$ and so lies in $H_{d-1}$). + +Therefore: the colour of an out spoke at depth $d$ equals the colour +of the corresponding face boundary edge at depth $d - 1$. + +This is the chain compatibility constraint: $T_d$'s out-spoke +projection $\pi_{\mathrm{out}}(T_d) \subseteq \{1, 2, 3\}^{|\text{out +spokes}|}$ corresponds, via the bijection $\{\text{out spokes of } +T_d\} \to \{\text{specific edges on face boundaries of } T_{d-1} +\text{'s}\}$, to a projection of $T_{d-1}$'s face-boundary colouring. + +Symmetrically: in spokes of $T_d$ ↔ specific face-boundary edges of +some $T_{d+1}$. + +\subsection*{Result inheritance via the partial-tire-dual identification} + +Because each cut tire \emph{is} a partial tire dual (up to graph +isomorphism), every counting/structural result for $D(T)$ in +\texttt{paper.tex} applies directly: + +\begin{itemize} + \item \textbf{Proposition 1.13 (edge--vertex coloring bijection):} + For spoke-only cut tires (face boundary a simple cycle of + length $n$, all $n$ spokes pendant), the number of proper + $3$-edge-colourings is $2^n + 2(-1)^n$. + \item \textbf{Definitions 1.15--1.17:} The face boundary plays the + role of $T'_{\mathrm{ann}}$; the spokes play the role of the + tire annular face connector $T'_{f'}$ and its inner/outer + spokes. + \item \textbf{Rainbow conjecture (\texttt{rainbow\_proof.tex}):} + For cut tires whose face boundary has the antipodal-chord SP + structure, the spoke projection support equals the + perms-per-face set $\mathcal{P}_m$, conditional on + Conjecture~1.5. + \item \textbf{Face-pair-connection structural description + (\texttt{k9\_surviving\_partitions.tex}):} Cut tires with + $r$ all-$3$ face structure admit exactly $2^r$ valid + ``induced partitions'' for König-lift purposes. +\end{itemize} + +These results provide quantitative bounds on $\pi_{\mathrm{in}}$ and +$\pi_{\mathrm{out}}$ at each layer. + +\subsection*{The chain pigeonhole step} + +For the chain $T_1 \to T_2 \to \cdots \to T_{d_{\max}}$ on side $i$: +\begin{itemize} + \item At the deep interior (depth $d_{\max}$), $T_{d_{\max}}$'s in + spokes either don't exist or terminate at the deepest + vertex. In the example $G'_1$ of Holton-McKay \#0, + $T_6$ has $2$ in spokes still going somewhere, but the chain + does eventually terminate. + \item Each $T_d$ constrains $T_{d-1}$ and $T_{d+1}$ via the + in/out-spoke correspondence with face boundary edges. + \item $\mathcal{R}_i = \pi_{\mathrm{out}}(\{T_1^{(i, f)}\})$ + as restricted by the chain of compatibility conditions + running from depth $1$ inward to $d_{\max}$. +\end{itemize} + +Chain pigeonhole says: if at each layer the realisable projection +contains enough structure (e.g.\ a full $S_3$-orbit), the chain +condition propagates through and yields a non-empty $\mathcal{R}_i$ +of substantial size. If both $\mathcal{R}_0$ and $\mathcal{R}_1$ +contain a common $S_3$-orbit, they intersect, contradicting that +$G'$ is a counterexample. + +\section*{What's actually new under the redefinition} + +\subsection*{1. Direct result transfer} + +Previously (with the original cut tire definition involving incident +edges in $G'_i$): we had to re-derive each result for cut tires, +because cut tires weren't formally $D(T)$. + +Now: each cut tire \emph{is} $D(T)$ for some virtual tire $T$, so +Prop 1.13, the rainbow proof, the König-lift framework, all apply +without re-derivation. + +\subsection*{2. Cubicity restored at face-boundary vertices} + +Each face-boundary vertex of a cut tire has degree $2$ (in $H_d$) + +$1$ pendant (in spoke or out spoke) $= 3$ in the cut tire. So +\emph{cut tires are cubic} (at face-boundary vertices), restoring +the cubic structure that the previous definition broke (where $H_d$ +was not cubic). + +This means classical cubic-graph results (König, Tait, chromatic +polynomial of $L(C_n)$, etc.) apply directly. + +\subsection*{3. The chain-defined structure is more rigid} + +Under the previous definition, ``the cut tire'' included variable +numbers of incident edges depending on $G'_i$'s local structure. +Under the new definition, the cut tire is determined by: +\begin{itemize} + \item The face $f$ (in $H_d$). + \item Which of $f$'s boundary vertices are degree-$2$ in $H_d$. + \item For each such vertex, whether its non-$H_d$ neighbour is at + depth $d - 1$ or $d + 1$. +\end{itemize} +This is finite, combinatorial, and exactly the data needed for +chain compatibility. + +\section*{What stays open} + +The fundamental obstacles \emph{don't} change: + +\subsection*{(a) Chain pigeonhole at each layer} + +The same open conjectures (rainbow, König-lift, 2-SAT solvability) +gate the chain argument. They now apply to cut tires directly +rather than via translation, but their statements are unchanged. + +\subsection*{(b) Chain well-definedness} + +\begin{itemize} + \item Each $H_d$ must have $\ge 1$ face. Empirically this holds in + the Holton-McKay example at all depths $1$--$7$, but no + general guarantee. + \item Some $H_d$ faces have length $2$ (just a single edge of + $H_d$ with two adjacent faces sharing it). These ``trivial'' + cut tires have empty face boundary cycle interior and + contribute very little structure. + \item Some boundary vertices have degree $> 2$ in $H_d$ + (cut-vertices or branch points of $H_d$); these don't get + pendants under the strict reading. 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}