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

coloring_nested_tire_graphs: add Birkhoff-Heesch reducibility note and dictionary to fiber view

Browse Source 
Summarises classical reducibility theory (Birkhoff 1913, Heesch
discharging, Appel-Haken, RSST) in modern notation, then maps it
onto the spoke-fiber decomposition: ring colorings ↔ spoke configs,
good/bad colorings ↔ realisable/unrealisable σ, D-reducibility ↔
chain-pigeonhole conductivity. Honest assessment: framework gives
vocabulary and a Sage-checkable template for small tires, but does
not give a uniform argument across tire sizes.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-26 01:17:17 -04:00
parent bfc68b7ec6
commit c5544497bd
4 changed files with 791 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/coloring_nested_tire_graphs/notes/birkhoff_heesch_reducibility.aux
+13
View File
@@ -0,0 +1,13 @@
\relax
\citation{thomas-update}
\citation{thomas-update}
\citation{bauerfeld-pds}
\citation{tutte-chromial}
\bibcite{birkhoff}{1}
\bibcite{heesch}{2}
\bibcite{appel-haken}{3}
\bibcite{rsst}{4}
\bibcite{thomas-update}{5}
\bibcite{tutte-chromial}{6}
\bibcite{bauerfeld-pds}{7}
\gdef \@abspage@last{7}
papers/coloring_nested_tire_graphs/notes/birkhoff_heesch_reducibility.log
+312
View File
@@ -0,0 +1,312 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 01:08
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
**birkhoff_heesch_reducibility.tex
(./birkhoff_heesch_reducibility.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/caption/caption.sty
Package: caption 2022/03/01 v3.6b Customizing captions (AR)
(/usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
Package: caption3 2022/03/17 v2.3b caption3 kernel (AR)
\caption@tempdima=\dimen158
\captionmargin=\dimen159
\caption@leftmargin=\dimen160
\caption@rightmargin=\dimen161
\caption@width=\dimen162
\caption@indent=\dimen163
\caption@parindent=\dimen164
\caption@hangindent=\dimen165
Package caption Info: Standard document class detected.
)
\c@caption@flags=\count274
\c@continuedfloat=\count275
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/tools/array.sty
Package: array 2021/10/04 v2.5f Tabular extension package (FMi)
\col@sep=\dimen166
\ar@mcellbox=\box52
\extrarowheight=\dimen167
\NC@list=\toks29
\extratabsurround=\skip55
\backup@length=\skip56
\ar@cellbox=\box53
)
(/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=\count276
\l__pdf_internal_box=\box54
)
(./birkhoff_heesch_reducibility.aux)
\openout1 = `birkhoff_heesch_reducibility.aux'.
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 19.
LaTeX Font Info: ... okay on input line 19.
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 19.
LaTeX Font Info: ... okay on input line 19.
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 19.
LaTeX Font Info: ... okay on input line 19.
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 19.
LaTeX Font Info: ... okay on input line 19.
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 19.
LaTeX Font Info: ... okay on input line 19.
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 19.
LaTeX Font Info: ... okay on input line 19.
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 19.
LaTeX Font Info: ... okay on input line 19.
(/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
\scratchcounter=\count277
\scratchdimen=\dimen168
\scratchbox=\box55
\nofMPsegments=\count278
\nofMParguments=\count279
\everyMPshowfont=\toks30
\MPscratchCnt=\count280
\MPscratchDim=\dimen169
\MPnumerator=\count281
\makeMPintoPDFobject=\count282
\everyMPtoPDFconversion=\toks31
) (/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)
Package caption Info: Begin \AtBeginDocument code.
Package caption Info: End \AtBeginDocument code.
LaTeX Font Info: Trying to load font information for U+msa on input line 20.
(/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 20.
(/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}] [2] [3]
[4] [5]
[6] [7] (./birkhoff_heesch_reducibility.aux) )
Here is how much of TeX's memory you used:
4568 strings out of 478268
73987 string characters out of 5846347
376623 words of memory out of 5000000
22749 multiletter control sequences out of 15000+600000
479245 words of font info for 64 fonts, out of 8000000 for 9000
1141 hyphenation exceptions out of 8191
55i,11n,63p,259b,221s stack positions out of 10000i,1000n,20000p,200000b,200000s
{/usr/local/texlive/2022/texmf-di
st/fonts/enc/dvips/cm-super/cm-super-ts1.enc}</usr/local/texlive/2022/texmf-dis
t/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist
/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/local/texlive/2022/texmf-dist/
fonts/type1/public/amsfonts/cm/cmbxti10.pfb></usr/local/texlive/2022/texmf-dist
/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/
fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/f
onts/type1/public/amsfonts/cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts
/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/t
ype1/public/amsfonts/cm/cmr12.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
e1/public/amsfonts/cm/cmr17.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pu
blic/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
lic/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publi
c/amsfonts/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
msfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
fonts/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
onts/symbols/msbm10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
cm-super/sfrm1095.pfb>
Output written on birkhoff_heesch_reducibility.pdf (7 pages, 252748 bytes).
PDF statistics:
124 PDF objects out of 1000 (max. 8388607)
76 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/birkhoff_heesch_reducibility.pdf
BIN
View File
Binary file not shown.
papers/coloring_nested_tire_graphs/notes/birkhoff_heesch_reducibility.tex
+466
View File
@@ -0,0 +1,466 @@
\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{graphicx}
\usepackage{geometry}
\usepackage{caption}
\usepackage{array}
\geometry{margin=1in}
\title{Birkhoff--Heesch reducibility and the fiber view\\
\large A dictionary between classical 4CT reducibility theory
and our spoke-fiber decomposition}
\author{}
\date{}
\newtheorem*{prop}{Proposition}
\newtheorem*{thm}{Theorem}
\newtheorem*{defn}{Definition}
\begin{document}
\maketitle
\section*{Purpose}
The conversation around the fiber decomposition note
(\texttt{fiber\_decomposition.tex}) flagged that the technique of
``split a coloring into a boundary configuration plus an
extension-counting fiber'' is exactly the framework of classical
reducibility theory for the Four-Color Theorem (4CT) --- developed by
Birkhoff (1913), then Heesch, Bernhart, Allaire, Swart, Appel--Haken,
and finally Robertson--Sanders--Seymour--Thomas (RSST, 1997). This
note (a) summarises the relevant pieces of that machinery in modern
notation following Thomas's survey~\cite{thomas-update}, then (b)
gives an explicit dictionary to our spoke-fiber language and an
honest assessment of what carries over.
\section*{Part I: classical reducibility, in modern notation}
\subsection*{The 4CT and its Tait dual}
\begin{thm}[4CT, vertex form]
Every plane graph $G$ has a proper $4$-vertex-coloring.
\end{thm}
\begin{thm}[Tait 1880, edge form]
Every cubic plane graph $H$ with no cut-edge has a proper $3$-edge-coloring.
This is equivalent to the 4CT.
\end{thm}
The equivalence goes through planar duality: a $4$-coloring of a
triangulation $G$, with colors $(0,0), (1,0), (0,1), (1,1)$ in
$\mathbb{Z}_2 \times \mathbb{Z}_2$, induces a $3$-edge-coloring of the
dual $G^*$ by assigning each dual edge the sum of the colors of the
two regions it separates. This is exactly the side our paper works
on: a triangulation $G$ has a cubic dual $G' \subseteq G^*$
(specifically: $G^*$ minus the unbounded face), and edge $3$-colorings
of $G'$ \emph{are} (up to handling of the outer face) $4$-colorings of $G$.
\subsection*{Birkhoff's minimum-counterexample setup (1913)}
The reducibility programme assumes for contradiction that there is a
\emph{minimum counterexample} to the 4CT --- a plane graph $T$ that is
not $4$-colorable but every smaller plane graph is. Birkhoff showed:
\begin{thm}[Birkhoff 1913]
Every minimum counterexample to the 4CT is an internally
$6$-connected triangulation.
\end{thm}
(``Internally $6$-connected'' means: removing fewer than $5$ vertices
keeps the graph connected, and removing any $5$ vertices leaves at
most a single isolated vertex.) All subsequent work assumes $T$ has
this form.
\subsection*{Configurations, rings, free completions}
A \emph{configuration} captures a small local piece of $T$
together with degree information.
\begin{defn}[Configuration, Thomas~\cite{thomas-update}]
A configuration is a pair $K = (H, \gamma)$ where $H$ is a
near-triangulation (one face is designated as ``special'', and every
other face is a triangle) and $\gamma : V(H) \to \mathbb{Z}_{\geq 5}$
satisfies:
\begin{enumerate}
\item[(i)] for interior vertices $v$ (not on the special face),
$\gamma(v) = \deg_H(v)$;
\item[(ii)] for boundary vertices $v$ (on the special face),
$\gamma(v) > \deg_H(v)$;
\item[(iii)] $\operatorname{ring-size}(K)
:= \sum_{v \text{ boundary}, H \setminus v \text{ connected}}
(\gamma(v) - \deg_H(v) - 1) \geq 2$.
\end{enumerate}
$K$ \emph{appears} in a triangulation $T$ if $H$ is an induced subgraph
of $T$, every non-special face of $H$ is a face of $T$, and $\gamma(v)$
equals the degree of $v$ in $T$ for every $v \in V(H)$.
\end{defn}
\begin{defn}[Free completion]
The \emph{free completion} of $K$ is the (essentially unique)
plane graph $S$ obtained from $H$ by adding a single cycle $R$ --- the
\emph{ring} of $K$ --- around the special face, plus the unique set of
edges making $S$ a triangulation of the disk bounded by $R$ in which
every vertex of $H$ has degree exactly $\gamma$. $R$ has length equal
to the ring-size of $K$.
\end{defn}
\subsection*{Color sets and reducibility}
Let $\mathcal{K}(R)$ be the set of all proper $4$-colorings of the ring
$R$ (often considered up to the $S_4$-action on colors, leaving roughly
$|\mathcal{K}(R)|/24$ orbits).
\begin{defn}[Good and bad colorings]
A coloring $\varphi \in \mathcal{K}(R)$ is \emph{good} if it extends
to a proper $4$-coloring of the free completion $S$, i.e.\ to all of
$H \cup R$. Otherwise it is \emph{bad}. Write
$\mathcal{C} \subseteq \mathcal{K}(R)$ for the set of good colorings.
\end{defn}
Now suppose $K$ appears in a minimum counterexample $T$, with free
completion $S \subseteq T$. Let $T' := T \setminus V(H)$ (the
``outside''); since $T$ is a minimum counterexample, $T'$ has a
$4$-coloring, which restricts to some
$\mathcal{C}' \subseteq \mathcal{K}(R)$. For $T$ to be a
counterexample,
\[
\mathcal{C}' \;\subseteq\; \mathcal{K}(R) \setminus \mathcal{C}
\]
(otherwise a coloring extending both into $S$ and into $T'$ gives a
$4$-coloring of all of $T$). The goal of reducibility is to show
$\mathcal{C}' = \emptyset$, contradicting that $T'$ is colorable.
\begin{defn}[Reducibility, classical taxonomy]
\hfill
\begin{itemize}
\item[A] $K$ is \textbf{A-reducible} if $\mathcal{C} = \mathcal{K}(R)$
(every ring coloring extends). A-reducibility immediately gives
$\mathcal{C}' = \emptyset$, but is too strong to hold for any
nontrivial configuration.
\item[D] $K$ is \textbf{D-reducible} if every bad coloring
$\varphi \in \mathcal{K}(R) \setminus \mathcal{C}$ can be
transformed --- via a sequence of Kempe-chain swaps on $T'$ ---
into a good coloring. D-reducibility is checkable by computer
on $\mathcal{K}(R)$ alone (no knowledge of $T'$ beyond that it
is $4$-colorable), because the Kempe-swap closure operation
can be applied at the level of ring colorings.
\item[C] $K$ is \textbf{C-reducible} if it is D-reducible \emph{after}
replacing $H$ by a smaller graph $H'$ (obtained from $H$ by
contracting up to four edges in RSST). This is strictly more
powerful than D-reducibility.
\end{itemize}
\end{defn}
Birkhoff's diamond (4 mutually adjacent pentagons surrounded by a
$6$-cycle ring) was the first nontrivial D-reducible configuration:
of the 31 ring colorings up to $S_4$-symmetry, 16 are good directly,
and the other 15 each admit a Kempe-swap to a good one.
\subsection*{Discharging and unavoidability}
Reducibility tells you that \emph{if} a good configuration appears in
$T$, then $T$ is not a counterexample. The other half of the proof
is to show that some good configuration \emph{must} appear:
\begin{defn}[Unavoidable set]
A set $\mathcal{U}$ of configurations is \emph{unavoidable} if every
internally $6$-connected triangulation contains at least one $K \in
\mathcal{U}$ as a subconfiguration.
\end{defn}
Heesch's \emph{discharging method} proves unavoidability:
\begin{enumerate}
\item Assign a \emph{charge} $\operatorname{ch}(v) := 6 - \deg(v)$ to each
vertex. By Euler's formula, $\sum_v \operatorname{ch}(v) = 12$ on
any triangulation of the sphere.
\item Define \emph{discharging rules} that redistribute charge
between vertices without changing the total.
\item Show that after discharging, no vertex carries positive charge
\emph{unless} it lies in (the neighborhood of) some
configuration in $\mathcal{U}$.
\item Since total charge is $12 > 0$, some positive-charge vertex
exists post-discharging, so some configuration in
$\mathcal{U}$ must appear.
\end{enumerate}
The two halves combine: 4CT $=$ ``every reducible configuration in
$\mathcal{U}$ blocks counterexamples'' $+$ ``$\mathcal{U}$ is unavoidable.''
\subsection*{The proofs}
\begin{itemize}
\item \textbf{Appel--Haken (1976/77).} $|\mathcal{U}| = 1936$
configurations (later reduced to $1482$); $487$ discharging
rules. Computer verification needed.
\item \textbf{RSST (1997).} $|\mathcal{U}| = 633$ configurations; $32$
discharging rules; quadratic-time $4$-coloring algorithm.
Still computer-assisted, but the unavoidability part was
written in a formal language and machine-checked.
\end{itemize}
For configurations of ring-size $r$, the number of colorings of $R$
modulo $S_4$ is roughly $3^{r-1}/3 + O(2^r)$ from the standard
chromatic-polynomial formula for $C_r$. RSST's largest ring-size is
$14$, with $\sim\!200{,}000$ ring colorings per configuration to
check.
\section*{Part II: dictionary to the fiber-decomposition view}
\subsection*{Side-of-the-duality conventions}
Our work is on the \emph{edge} side of Tait: we color edges of $G'$
with $3$ colors, where $G'$ is essentially the dual of a triangulation
$G$. Birkhoff--Heesch is on the \emph{vertex} side of $G$. By Tait's
theorem these are equivalent in principle, but the shape of a
``configuration boundary'' looks different on the two sides:
\begin{center}
\begin{tabular}{>{\raggedright\arraybackslash}p{0.36\textwidth}|>{\raggedright\arraybackslash}p{0.55\textwidth}}
\textbf{Vertex side (Birkhoff--Heesch)} & \textbf{Edge side (us)} \\ \hline
Triangulation $T$ & Cubic dual $G'$ \\
Configuration $K = (H, \gamma)$ in $T$ & Tire annular face connector
$T'_{f'} \subseteq G'$ \\
Inner-face triangles of $H$ & Vertices of $V(f')$ (dual to annular
faces of the tire) \\
Ring $R$ (a cycle around $H$) & Boundary of $T'_{f'}$, consisting of
spoke edges $E_S$ \emph{plus} (in the multi-tire chain) the cycle
$V(f')$ itself \\
Ring coloring $\varphi : V(R) \to \{1,2,3,4\}$ & Spoke configuration
$\sigma : E_S \to \{1,2,3\}$ \\
$\mathcal{K}(R)$ = all proper $4$-colorings of $R$ &
$\Sigma := $ all spoke configurations $\sigma$ (whether
realisable or not) \\
$\mathcal{C}$ = good (extendable to $H \cup R$) ring colorings &
Realisable spoke configurations:
$\{\sigma : N(T'_{f'}; \sigma) > 0\}$ \\
$\mathcal{K}(R) \setminus \mathcal{C}$ = bad ring colorings &
Unrealisable $\sigma$: the boundary of $T'_{f'}$'s reachability \\
Fiber over a ring coloring: \# extensions of $\varphi$ to $H \cup R$ &
Fiber count $N(T'_{f'}; \sigma)$ \\
Kempe-chain swap on $T \setminus V(H)$ &
Tait-Kempe swap on $G' \setminus V(T'_{f'})$ (two-color
alternating-edge swap)
\end{tabular}
\end{center}
\subsection*{The fiber identity, in Birkhoff language}
Our identity
$P_e(T'_{f'}, 3) = \sum_\sigma N(T'_{f'}; \sigma)$
is literally Birkhoff's ``good colorings'' decomposition on the edge
side, with \emph{the same scaffold}: count interior extensions
fibered over boundary state. The classical literature does not
emphasise this identity per se (it's the trivial sum), but every
reducibility analysis is built around the fiber distribution
$\{N(T'_{f'}; \sigma)\}_\sigma$.
\subsection*{What ``reducible'' means in our world}
Translating definitions of A-, D-, C-reducibility through the
dictionary:
\begin{itemize}
\item \textbf{Edge-A-reducibility of $T'_{f'}$}: every spoke
configuration $\sigma \in \Sigma$ is realisable. This would
mean $T'_{f'}$ supports an edge $3$-coloring extension from
\emph{any} boundary input --- a very strong locally-flexible
property.
\item \textbf{Edge-D-reducibility of $T'_{f'}$}: every unrealisable
$\sigma$ can be Tait-Kempe-swapped (in the outside graph
$G' \setminus T'_{f'}$) to a realisable one. This is the
natural form of the conductivity step from our chain-pigeonhole
sketch.
\item \textbf{Edge-C-reducibility of $T'_{f'}$}: D-reducibility
after replacing $T'_{f'}$ by a smaller subgraph (e.g.\
contracting some spoke edges or merging adjacent faces).
\end{itemize}
\subsection*{The chain-pigeonhole step as a reducibility statement}
Our conductivity step --- ``the middle tire $T_B$'s
$\phi_B : \mathcal{P}_{AB} \to 2^{\mathcal{P}_{BC}}$ takes any input
to a $>\!|\mathcal{P}_{BC}|/2$-sized output'' --- is the
\emph{quantitative} analogue of D-reducibility, but applied to
\emph{composition} of configurations rather than reduction. In
particular:
\begin{itemize}
\item Birkhoff--Heesch D-reducibility says ``bad inputs to a single
configuration can be Kempe-moved to good ones.''
\item Our chain-pigeonhole says ``the bad inputs to the inner tire
and the bad inputs to the outer tire together don't cover
$\mathcal{P}_\gamma$, so something good remains.''
\end{itemize}
These are not the same statement, but they are about the same data
$\mathcal{C}$ vs.\ $\mathcal{K} \setminus \mathcal{C}$ on the shared
cycle.
\section*{Part III: does the machinery apply?}
\subsection*{What carries over cleanly}
\begin{enumerate}
\item \textbf{Vocabulary and scaffolding.} Configurations,
rings, free completions, color sets, fiber-of-extensions are
all there; the menagerie/fiber notes are using the same
objects.
\item \textbf{Kempe-chain machinery.} Tait-Kempe chains (two-color
alternating edge paths in $G'$) are a real, well-developed
tool. Any conductivity argument we want is fundamentally
Tait-Kempe in flavour.
\item \textbf{Computer-verifiability for small tires.} Just as
D-reducibility of an individual configuration is verified by
enumerating $\sim\!200{,}000$ ring colorings, the fiber
distribution $\{N(T'_{f'}; \sigma)\}$ for a fixed small tire
is a finite Sage computation. We can test conductivity
empirically before trying to prove it.
\end{enumerate}
\subsection*{What does \emph{not} carry over straightforwardly}
\begin{enumerate}
\item \textbf{The reducibility scale.} Classical reducibility was
only ever practical for ring-size $\leq 14$ because the number
of ring colorings explodes. Our tires can have arbitrarily
large boundary cycles (in a multi-layer triangulation, the
annular ring is whatever the level structure gives), so the
\emph{single-configuration} reducibility approach hits the
same wall as A\&H/RSST: only small tires are tractable
directly.
\item \textbf{Compositionality.} Birkhoff--Heesch operates on
\emph{one} configuration at a time. Our chain
pigeonhole/nesting is fundamentally about \emph{composing}
configurations along shared boundaries. This is a structural
feature classical reducibility does \emph{not} engage with ---
their unavoidability argument (discharging) replaces it. If
we want to prove a statement of the form ``every nested chain
of tires admits a global $4$-coloring,'' Birkhoff--Heesch does
not directly give us the tool; we need either a transfer-matrix
/ monotonicity argument across nestings or a structural
result about how the realisable supports behave under
composition.
\item \textbf{Unavoidability is automatic / different.} In 4CT,
unavoidability is a separate hard problem solved by
discharging. In our setup the tire decomposition of
$G$ is given by the level structure (Bauerfeld
\cite{bauerfeld-pds}), so there is no analogous unavoidability
question --- the tires are already there. This means
\emph{half} of the classical apparatus (discharging) is not
the point of contact; the contact is purely on the
reducibility/color-set side.
\item \textbf{Tait correspondence is global, not local.} Edge
$3$-coloring of $G'$ globally encodes a vertex $4$-coloring of
$G$, but the local correspondence between an edge coloring of
$T'_{f'}$ and a vertex coloring of the corresponding piece of
$G$ is subtle: edge swaps on the dual side do not always
correspond to single Kempe chains on the primal side, and the
``ring'' as a vertex cycle in $G$ may differ in length and
structure from the ``boundary'' of $T'_{f'}$ on the edge side.
Any time we want to import a vertex-side Kempe argument we
will have to do the bookkeeping carefully.
\end{enumerate}
\subsection*{Assessment}
\textbf{Yes, the Birkhoff--Heesch framework applies, but only as a
language and as a verification tool for small instances.} It does
\emph{not} hand us a proof of the nesting/chain-pigeonhole conjecture
for free, because:
\begin{itemize}
\item Classical reducibility is single-configuration; our argument
is multi-configuration / compositional.
\item Classical reducibility's quantitative input is exhaustive
enumeration of ring colorings up to size $14$; we want
statements uniform over arbitrarily large tires.
\end{itemize}
What it \emph{does} hand us:
\begin{itemize}
\item A precise vocabulary --- good/bad colorings, free completion,
D-/C-reducibility --- so that the conductivity step can be
stated in established terms.
\item A concrete computational template: for any specific tire,
compute $\{N(T'_{f'}; \sigma)\}_\sigma$ in Sage and check
whether the realisable support $\mathcal{C}$ is large enough
that two adjacent tires must overlap. This is the
analogue of mechanical D-reducibility checking.
\item Strong evidence about the difficulty: nothing in 80+ years of
reducibility work has reduced 4CT to a structural argument
across all configuration sizes. If our chain argument
succeeds, it will be \emph{because tires are a more
structured class than arbitrary configurations}, not because
the reducibility apparatus suddenly gives uniform results.
\end{itemize}
\subsection*{Concrete next steps}
\begin{enumerate}
\item \textbf{Pick a small tire family} ($B_{\mathrm{in}}$ a $k$-cycle
for $k \in \{3, 4, 5, 6\}$, $B_{\mathrm{out}}$ a small cycle,
no $O$-chords) and compute the full fiber distribution
$\{N(T'_{f'}; \sigma)\}_\sigma$ in Sage. This gives the realisable
support $\mathcal{C}$ and, in classical terms, tests whether
small tires are A-reducible (full support), D-reducible
(Kempe-recoverable), or neither.
\item \textbf{Check support overlap for adjacent tires.} Given two
small tires sharing a cycle $\gamma$, do the realisable
supports $\mathcal{C}_{\mathrm{out}}, \mathcal{C}_{\mathrm{in}}
\subseteq \mathcal{K}(\gamma)$ always intersect? This is the
empirical version of the chain-pigeonhole step. If they
sometimes miss, the simple form of the argument fails and
we need a Kempe-chain (D-reducibility-style) escape route.
\item \textbf{Look up monosystems / Tutte.} Tutte
\cite{tutte-chromial} formulated 4CT-adjacent counting
problems in terms of his \emph{chromial}; this is the closest
existing transfer-matrix-style framing of these color sets,
and may give a cleaner composition rule than the raw fiber
sum.
\item \textbf{Look up Sokal / Chang--Shrock} on chromatic
polynomials of strip graphs. They use transfer matrices for
infinite families of small-ring configurations, which is
structurally similar to nested tires with fixed ring size.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{birkhoff}
G.~D.~Birkhoff,
\emph{The reducibility of maps},
Amer.\ J.\ Math.\ \textbf{35} (1913), 115--128.
\bibitem{heesch}
H.~Heesch,
\emph{Untersuchungen zum Vierfarbenproblem},
B.~I.\ Hochschulskripten 810/810a/810b, Bibliographisches Institut,
Mannheim, 1969.
\bibitem{appel-haken}
K.~Appel and W.~Haken,
\emph{Every planar map is four colorable. Part I: Discharging};
\emph{Part II: Reducibility} (with J.~Koch),
Illinois J.\ Math.\ \textbf{21} (1977), 429--567.
\bibitem{rsst}
N.~Robertson, D.~P.~Sanders, P.~Seymour, R.~Thomas,
\emph{The Four-Colour Theorem},
J.\ Combin.\ Theory Ser.\ B \textbf{70} (1997), 2--44.
\bibitem{thomas-update}
R.~Thomas,
\emph{An update on the Four-Color Theorem},
Notices Amer.\ Math.\ Soc.\ \textbf{45} (1998), 848--859.
\bibitem{tutte-chromial}
W.~T.~Tutte,
\emph{Chromials},
in: \emph{Hypergraph Seminar} (C.~Berge, D.~Ray-Chaudhuri, eds.),
Lecture Notes in Math.\ 411, Springer, 1974, pp.~243--266.
\bibitem{bauerfeld-pds}
E.~Bauerfeld,
\emph{Plane Depth Sequencing},
manuscript (math-research repository), 2026.
\end{thebibliography}
\end{document}