didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

coloring_nested_tire_graphs: examine chain pigeonhole on cut tires

Browse Source 
New note cut_tire_chain_pigeonhole.tex (3 pages) walking through the
chain pigeonhole argument applied to the cut-tire framework:

Setup: minimum counterexample G' to 4CT, 6-edge cut splits into
G'_0, G'_1, depth labelling gives chains of cut tires on each side.

Argument shape:
  (1) Minimality ⇒ both G'_i have proper 3-edge-colorings.
  (2) Restrict to depth-0 pendants → boundary configurations σ_0, σ_1.
  (3) Glue iff σ_0 = σ_1; counterexample ⇒ no such matching.
  (4) Layered description: each cut tire has inner/outer projection
      constraints; adjacent tires share layers (outer of T_d =
      inner of T_{d+1}).
  (5) Chain pigeonhole: if each π_in is "large enough,"
      R_0 ∩ R_1 ≠ ∅, contradiction.

What this needs to be a proof:
  (a) Chain well-definedness: each H_d has ≥ 1 face, adjacencies
      clean, no degenerate cases.
  (b) Quantitative chain pigeonhole at each layer (= the rainbow
      conjecture or König-lift conjecture from existing notes).
  (c) Cut-tire-specific issues: H_d not cubic, face boundaries may
      not be simple cycles, transfer of primal-tire results requires
      verification.

Empirical chain on G'_1 of Holton-McKay #0: chain length 7,
irregular face structure across depths.  No counterexample yet but
no proof either.

Net assessment: structurally sound reformulation, but inherits all
open conjectures from the existing approaches plus new technical
issues.

Concrete next step: extend tire_fiber_step2-style pairwise compatibility
check to the cut-tire setting and see if R_0 ∩ R_1 = ∅ empirically
on the Holton-McKay graphs.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-26 15:37:14 -04:00
parent 3172371fb8
commit 96102e6d03
4 changed files with 575 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.aux
+8
View File
@@ -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 \@abspage@last{4}
papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.log
+315
View File
@@ -0,0 +1,315 @@
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
restricted \write18 enabled.
%&-line parsing enabled.
**cut_tire_chain_pigeonhole.tex
(./cut_tire_chain_pigeonhole.tex
LaTeX2e <2021-11-15> patch level 1
L3 programming layer <2022-02-24>
(/usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
Document Class: article 2021/10/04 v1.4n Standard LaTeX document class
(/usr/local/texlive/2022/texmf-dist/tex/latex/base/size11.clo
File: size11.clo 2021/10/04 v1.4n Standard LaTeX file (size option)
)
\c@part=\count185
\c@section=\count186
\c@subsection=\count187
\c@subsubsection=\count188
\c@paragraph=\count189
\c@subparagraph=\count190
\c@figure=\count191
\c@table=\count192
\abovecaptionskip=\skip47
\belowcaptionskip=\skip48
\bibindent=\dimen138
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
Package: amsmath 2021/10/15 v2.17l AMS math features
\@mathmargin=\skip49
For additional information on amsmath, use the `?' option.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
Package: amstext 2021/08/26 v2.01 AMS text
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
File: amsgen.sty 1999/11/30 v2.0 generic functions
\@emptytoks=\toks16
\ex@=\dimen139
))
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
Package: amsbsy 1999/11/29 v1.2d Bold Symbols
\pmbraise@=\dimen140
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
Package: amsopn 2021/08/26 v2.02 operator names
)
\inf@bad=\count193
LaTeX Info: Redefining \frac on input line 234.
\uproot@=\count194
\leftroot@=\count195
LaTeX Info: Redefining \overline on input line 399.
\classnum@=\count196
\DOTSCASE@=\count197
LaTeX Info: Redefining \ldots on input line 496.
LaTeX Info: Redefining \dots on input line 499.
LaTeX Info: Redefining \cdots on input line 620.
\Mathstrutbox@=\box50
\strutbox@=\box51
\big@size=\dimen141
LaTeX Font Info: Redeclaring font encoding OML on input line 743.
LaTeX Font Info: Redeclaring font encoding OMS on input line 744.
\macc@depth=\count198
\c@MaxMatrixCols=\count199
\dotsspace@=\muskip16
\c@parentequation=\count266
\dspbrk@lvl=\count267
\tag@help=\toks17
\row@=\count268
\column@=\count269
\maxfields@=\count270
\andhelp@=\toks18
\eqnshift@=\dimen142
\alignsep@=\dimen143
\tagshift@=\dimen144
\tagwidth@=\dimen145
\totwidth@=\dimen146
\lineht@=\dimen147
\@envbody=\toks19
\multlinegap=\skip50
\multlinetaggap=\skip51
\mathdisplay@stack=\toks20
LaTeX Info: Redefining \[ on input line 2938.
LaTeX Info: Redefining \] on input line 2939.
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
Package: amssymb 2013/01/14 v3.01 AMS font symbols
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
\symAMSa=\mathgroup4
\symAMSb=\mathgroup5
LaTeX Font Info: Redeclaring math symbol \hbar on input line 98.
LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
(Font) U/euf/m/n --> U/euf/b/n on input line 106.
))
(/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
Package: amsthm 2020/05/29 v2.20.6
\thm@style=\toks21
\thm@bodyfont=\toks22
\thm@headfont=\toks23
\thm@notefont=\toks24
\thm@headpunct=\toks25
\thm@preskip=\skip52
\thm@postskip=\skip53
\thm@headsep=\skip54
\dth@everypar=\toks26
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
Package: keyval 2014/10/28 v1.15 key=value parser (DPC)
\KV@toks@=\toks27
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
Package: trig 2021/08/11 v1.11 sin cos tan (DPC)
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
)
Package graphics Info: Driver file: pdftex.def on input line 107.
(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex
))
\Gin@req@height=\dimen148
\Gin@req@width=\dimen149
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
Package: geometry 2020/01/02 v5.9 Page Geometry
(/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
Package: ifvtex 2019/10/25 v1.7 ifvtex legacy package. Use iftex instead.
(/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
Package: iftex 2022/02/03 v1.0f TeX engine tests
))
\Gm@cnth=\count271
\Gm@cntv=\count272
\c@Gm@tempcnt=\count273
\Gm@bindingoffset=\dimen150
\Gm@wd@mp=\dimen151
\Gm@odd@mp=\dimen152
\Gm@even@mp=\dimen153
\Gm@layoutwidth=\dimen154
\Gm@layoutheight=\dimen155
\Gm@layouthoffset=\dimen156
\Gm@layoutvoffset=\dimen157
\Gm@dimlist=\toks28
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
Package: booktabs 2020/01/12 v1.61803398 Publication quality tables
\heavyrulewidth=\dimen158
\lightrulewidth=\dimen159
\cmidrulewidth=\dimen160
\belowrulesep=\dimen161
\belowbottomsep=\dimen162
\aboverulesep=\dimen163
\abovetopsep=\dimen164
\cmidrulesep=\dimen165
\cmidrulekern=\dimen166
\defaultaddspace=\dimen167
\@cmidla=\count274
\@cmidlb=\count275
\@aboverulesep=\dimen168
\@belowrulesep=\dimen169
\@thisruleclass=\count276
\@lastruleclass=\count277
\@thisrulewidth=\dimen170
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
\l__color_backend_stack_int=\count278
\l__pdf_internal_box=\box52
)
(./cut_tire_chain_pigeonhole.aux)
\openout1 = `cut_tire_chain_pigeonhole.aux'.
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 18.
LaTeX Font Info: ... okay on input line 18.
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 18.
LaTeX Font Info: ... okay on input line 18.
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 18.
LaTeX Font Info: ... okay on input line 18.
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 18.
LaTeX Font Info: ... okay on input line 18.
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 18.
LaTeX Font Info: ... okay on input line 18.
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 18.
LaTeX Font Info: ... okay on input line 18.
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 18.
LaTeX Font Info: ... okay on input line 18.
(/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
\scratchcounter=\count279
\scratchdimen=\dimen171
\scratchbox=\box53
\nofMPsegments=\count280
\nofMParguments=\count281
\everyMPshowfont=\toks29
\MPscratchCnt=\count282
\MPscratchDim=\dimen172
\MPnumerator=\count283
\makeMPintoPDFobject=\count284
\everyMPtoPDFconversion=\toks30
) (/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4
85.
(/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
e
))
*geometry* driver: auto-detecting
*geometry* detected driver: pdftex
*geometry* verbose mode - [ preamble ] result:
* driver: pdftex
* paper: <default>
* layout: <same size as paper>
* layoutoffset:(h,v)=(0.0pt,0.0pt)
* modes:
* h-part:(L,W,R)=(72.26999pt, 469.75502pt, 72.26999pt)
* v-part:(T,H,B)=(72.26999pt, 650.43001pt, 72.26999pt)
* \paperwidth=614.295pt
* \paperheight=794.96999pt
* \textwidth=469.75502pt
* \textheight=650.43001pt
* \oddsidemargin=0.0pt
* \evensidemargin=0.0pt
* \topmargin=-37.0pt
* \headheight=12.0pt
* \headsep=25.0pt
* \topskip=11.0pt
* \footskip=30.0pt
* \marginparwidth=59.0pt
* \marginparsep=10.0pt
* \columnsep=10.0pt
* \skip\footins=10.0pt plus 4.0pt minus 2.0pt
* \hoffset=0.0pt
* \voffset=0.0pt
* \mag=1000
* \@twocolumnfalse
* \@twosidefalse
* \@mparswitchfalse
* \@reversemarginfalse
* (1in=72.27pt=25.4mm, 1cm=28.453pt)
LaTeX Font Info: Trying to load font information for U+msa on input line 19.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
)
LaTeX Font Info: Trying to load font information for U+msb on input line 19.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
) [1
{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
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 140.
Overfull \hbox (0.83797pt too wide) in paragraph at lines 147--153
[]\OT1/cmr/bx/n/10.95 K^^?onig-lift con-jec-ture (\OT1/cmtt/m/n/10.95 worst[]ca
se[]proof[]sketch.tex\OT1/cmr/bx/n/10.95 , Conj \OT1/cmr/bx/it/10.95 t2-induces
-partition\OT1/cmr/bx/n/10.95 ): \OT1/cmr/m/n/10.95 ad-
[]
[2]
Overfull \hbox (3.26193pt too wide) in paragraph at lines 224--228
[]\OT1/cmr/m/n/10.95 Inherits all the open con-jec-tures from \OT1/cmtt/m/n/10.
95 rainbow[]proof.tex \OT1/cmr/m/n/10.95 and \OT1/cmtt/m/n/10.95 worst[]case[]p
roof[]sketch.tex
[]
[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}</usr/local/texlive/2022/texmf-dist/f
onts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fo
nts/type1/public/amsfonts/cm/cmbx12.pfb></usr/local/texlive/2022/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmbxti10.pfb></usr/local/texlive/2022/texmf-dist/fo
nts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts
/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/t
ype1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
e1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
/public/amsfonts/cm/cmr17.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publ
ic/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
msfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
fonts/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
nts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/cm-supe
r/sfrm1095.pfb>
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)
papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.pdf
BIN
View File
Binary file not shown.
papers/coloring_nested_tire_graphs/notes/cut_tire_chain_pigeonhole.tex
+252
View File
@@ -0,0 +1,252 @@
\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}