diff --git a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux index 4bed7a9..f9dcfe2 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux +++ b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux @@ -4,9 +4,13 @@ \@writefile{toc}{\contentsline {paragraph}{Visualization.}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Relation to the existing tire framework.}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Example on $G'_1$ (Holton-McKay \#0, $V \setminus S$ half).}{3}{}\protected@file@percent } -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{5}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Justification of the per-tire half.}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Justification of the chain half.}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{What this would (and would not) prove.}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Status.}{4}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{6}{}\protected@file@percent } \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} -\newlabel{fig:cut-depth-label}{{1}{5}} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$. In each panel, blue solid edges form the face boundary at depth $d$; red dashed edges are inner spokes (depth $d - 1$); green dashed edges are outer spokes (depth $d + 1$). Vertices on the face boundary are highlighted; the rest of $G'_1$ is faded.\relax }}{6}{}\protected@file@percent } -\newlabel{fig:cut-tire}{{2}{6}} -\gdef \@abspage@last{6} +\newlabel{fig:cut-depth-label}{{1}{6}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$. In each panel, blue solid edges form the face boundary at depth $d$; red dashed edges are inner spokes (depth $d - 1$); green dashed edges are outer spokes (depth $d + 1$). Vertices on the face boundary are highlighted; the rest of $G'_1$ is faded.\relax }}{7}{}\protected@file@percent } +\newlabel{fig:cut-tire}{{2}{7}} +\gdef \@abspage@last{7} diff --git a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.log b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.log index 31b8ae7..d4c885a 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.log +++ b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.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 15:56 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 16:15 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -311,19 +311,40 @@ Package pdftex.def Info: fig_cut_tire.png used on input line 196. LaTeX Warning: `h' float specifier changed to `ht'. -Overfull \hbox (46.78696pt too wide) in paragraph at lines 223--230 +Overfull \hbox (65.5249pt too wide) in paragraph at lines 223--231 +\OT1/cmr/m/n/10.95 The rain-bow con-jec-ture and K^^?onig-lift con-jec-ture (fr +om \OT1/cmtt/m/n/10.95 rainbow[]proof.tex \OT1/cmr/m/n/10.95 and \OT1/cmtt/m/n/ +10.95 worst[]case[]proof[]sketch.tex\OT1/cmr/m/n/10.95 ) + [] + + +Overfull \hbox (0.56282pt too wide) in paragraph at lines 267--278 + \OT1/cmr/bx/n/10.95 Jus-ti-fi-ca-tion of the per-tire half.[] \OT1/cmr/m/n/10. +95 The per-tire half is es-sen-tially Propo-si-tion 1.13 of \OT1/cmtt/m/n/10.95 + paper.tex\OT1/cmr/m/n/10.95 : + [] + +[3] +Overfull \hbox (115.9405pt too wide) in paragraph at lines 300--307 + \OT1/cmr/bx/n/10.95 Sta-tus.[] \OT1/cmr/m/n/10.95 Em-pir-i-cally true for ev-e +ry cut tire in the Holton-McKay #0 ex-am-ple (tested in \OT1/cmtt/m/n/10.95 cut +[]tire[]conjecture[]tests.tex\OT1/cmr/m/n/10.95 ). + [] + + +Overfull \hbox (46.78696pt too wide) in paragraph at lines 310--317 \OT1/cmr/m/n/10.95 The pro-ce-dure mir-rors the $4$CT cut-and-reglue scheme (\O T1/cmtt/m/n/10.95 rainbow[]proof.tex\OT1/cmr/m/n/10.95 , \OT1/cmtt/m/n/10.95 wo rst[]case[]proof[]sketch.tex\OT1/cmr/m/n/10.95 , [] -[3] [4] [5 <./fig_cut_depth_label.png>] [6 <./fig_cut_tire.png>] +[4] [5] [6 <./fig_cut_depth_label.png>] [7 <./fig_cut_tire.png>] (./cut_depth_label.aux) ) Here is how much of TeX's memory you used: - 4592 strings out of 478268 - 74512 string characters out of 5846347 - 375437 words of memory out of 5000000 - 22771 multiletter control sequences out of 15000+600000 + 4593 strings out of 478268 + 74520 string characters out of 5846347 + 376437 words of memory out of 5000000 + 22772 multiletter control sequences out of 15000+600000 479342 words of font info for 67 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 55i,8n,63p,656b,312s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -343,10 +364,10 @@ i10.pfb> -Output written on cut_depth_label.pdf (6 pages, 509609 bytes). +Output written on cut_depth_label.pdf (7 pages, 519218 bytes). PDF statistics: - 99 PDF objects out of 1000 (max. 8388607) - 58 compressed objects within 1 object stream + 104 PDF objects out of 1000 (max. 8388607) + 62 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 11 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.pdf b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.pdf index a1115af..8f29350 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.pdf and b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex index 250196d..841007a 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex +++ b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex @@ -218,6 +218,93 @@ under the redefinition each contribute exactly one in/out spoke): tire in this chain. \end{itemize} +\section*{A looser chain pigeonhole hypothesis} + +The rainbow conjecture and K\"onig-lift conjecture (from +\texttt{rainbow\_proof.tex} and +\texttt{worst\_case\_proof\_sketch.tex}) are statements about +specific structured face boundaries (antipodal-chord SP). Generic +cut tires in our example do not have those specific structures, so +those conjectures don't apply directly. Empirically +(\texttt{cut\_tire\_conjecture\_tests.tex}), cut-tire projections +are nonetheless large and well-behaved: + +\begin{itemize} + \item Every cut tire's joint projection $\pi(T) := + \{(\sigma_{\mathrm{out}}, \sigma_{\mathrm{in}}) : \chi + \text{ proper}\}$ is $S_3$-closed (universal, follows from + color symmetry of proper edge $3$-colouring). + \item All $S_3$-orbits in $\pi(T)$ have size exactly $3$ (the + constant-colour orbit, when present) or $6$ (the generic + orbit using all $3$ colours); no size-$2$ stabilizer orbits + occur. + \item Non-trivial cut tires have $|\pi(T)| \gg 6$: e.g.\ at + depth~$2$ the projection has size $126 = 21 \cdot 6$ + ($21$ full $S_3$-orbits), and at depth~$6$ size $93 = 1 + \cdot 3 + 15 \cdot 6$. +\end{itemize} + +This motivates a much weaker conjecture than the rainbow/K\"onig +ones --- one that does not require special face-boundary structure +and may be easier to prove: + +\begin{quote} +\textbf{Loose chain pigeonhole conjecture (cut tires).} Let +$T = T_d^{(i, f)}$ be a non-trivial cut tire (one with at least one +in or out spoke). Then the joint projection $\pi(T)$ is non-empty, +$S_3$-closed, and contains at least one full $S_3$-orbit (i.e.\ +$|\pi(T)| \geq 6$, with equality only in the degenerate case where +$\pi(T)$ is a single non-constant $S_3$-orbit). + +Consequently, for the chain $T_1^{(i)} \to T_2^{(i)} \to \cdots$ on +each side of a cut, the realisable cut-configuration set +$\mathcal{R}_i$ contains a full $S_3$-orbit, and the intersection +$\mathcal{R}_0 \cap \mathcal{R}_1$ contains a common $S_3$-orbit +(of size $6$), forcing $\mathcal{R}_0 \cap \mathcal{R}_1 \ne +\emptyset$. +\end{quote} + +\paragraph{Justification of the per-tire half.} The per-tire half +is essentially Proposition~1.13 of \texttt{paper.tex}: a spoke-only +cut tire (face boundary a simple cycle of length $n$) has $2^n + +2(-1)^n$ proper edge $3$-colourings, which is $\ge 6$ whenever +$n \ge 3$. By $S_3$-closure, the orbit count is at least $1$, +hence $\ge 6$ distinct projections (modulo the size-$3$ constant +orbit edge case). For cut tires with non-simple face boundary +(e.g.\ $\theta(1, p, q)$-shape) similar lower bounds follow from +the chromatic polynomial of the corresponding $D(T)$ structure (see +the menagerie note for explicit closed forms when chord matchings +are present). + +\paragraph{Justification of the chain half.} Chain composition +preserves $S_3$-symmetry (each step is $S_3$-equivariant), so +$\mathcal{R}_i$ inherits $S_3$-closure from the per-tire projection. +The non-trivial claim is that the orbit structure \emph{survives} +composition: an $S_3$-orbit in $\pi(T_d)$ projects to some +non-empty $S_3$-closed subset on the out-spoke side, and this +propagates layer-by-layer. + +\paragraph{What this would (and would not) prove.} If the loose +conjecture holds, chain pigeonhole at the cut gives $\mathcal{R}_0 +\cap \mathcal{R}_1 \neq \emptyset$, which (under the minimum- +counterexample reduction) contradicts $G'$ being a 4CT +counterexample. This is the same final conclusion as the +rainbow/K\"onig route, with weaker hypotheses but stronger +empirical support. + +It would \emph{not} prove the rainbow conjecture itself (which is a +strict structural claim about the antipodal-chord SP case), nor +the K\"onig-lift conjecture's specific construction; it would only +recover the bottom-line chain pigeonhole result. + +\paragraph{Status.} Empirically true for every cut tire in the +Holton-McKay \#0 example (tested in +\texttt{cut\_tire\_conjecture\_tests.tex}). No general proof. +The per-tire half is essentially provable via Prop~1.13 plus +$S_3$-equivariance. The chain half is the genuinely open question +--- it requires showing that the per-tire $S_3$-orbit structure +composes coherently through the depth chain. + \section*{Connection to chain pigeonhole / 4CT reducibility} The procedure mirrors the $4$CT cut-and-reglue scheme