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..15389c7 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.aux @@ -0,0 +1,8 @@ +\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 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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}