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 deleted file mode 100644 index 15389c7..0000000 --- a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.aux +++ /dev/null @@ -1,8 +0,0 @@ -\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} 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 deleted file mode 100644 index a56292a..0000000 --- a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.log +++ /dev/null @@ -1,315 +0,0 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 15:37 -entering extended mode - 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[] - -[3] [4] (./cut_tire_chain_pigeonhole.aux) ) -Here is how much of TeX's memory you used: - 3257 strings out of 478268 - 48521 string characters out of 5846347 - 350622 words of memory out of 5000000 - 21443 multiletter control sequences out of 15000+600000 - 479884 words of font info for 69 fonts, out of 8000000 for 9000 - 1141 hyphenation exceptions out of 8191 - 55i,7n,62p,242b,218s stack positions out of 10000i,1000n,20000p,200000b,200000s -{/usr/local/texlive/2022/texmf-dist/ -fonts/enc/dvips/cm-super/cm-super-ts1.enc} -Output written on cut_tire_chain_pigeonhole.pdf (4 pages, 211212 bytes). -PDF statistics: - 103 PDF objects out of 1000 (max. 8388607) - 62 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 deleted file mode 100644 index 197001f..0000000 Binary files a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf and /dev/null 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 deleted file mode 100644 index 88c2551..0000000 --- a/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex +++ /dev/null @@ -1,252 +0,0 @@ -\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} diff --git a/papers/coloring_nested_tire_graphs/notes/induced_partition_findings.log b/papers/coloring_nested_tire_graphs/notes/induced_partition_findings.log index 8db2f40..40e5fed 100644 --- a/papers/coloring_nested_tire_graphs/notes/induced_partition_findings.log +++ b/papers/coloring_nested_tire_graphs/notes/induced_partition_findings.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 11:33 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 11:44 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -276,7 +276,7 @@ S/cmsy/m/n/10.95 g\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 f\OT1/cmr/m/n/10.95 {/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2] 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 171. +(Font) Font shape `OT1/cmtt/m/n' tried instead on input line 168. 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