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

dual_decomposition: split iterated reduction into companion paper

Browse Source 
- Move the iterated-reduction algorithm, its two structural lemmas
  (exactly-one-match, all-distinct-exists), and the n=14 trace figure
  into a new companion paper at
  papers/dual_decomposition_iterated_reduction/. Figures and figure
  scripts moved via git mv (history preserved).
- In the main paper, Section 3 ("An iterated reduction") becomes
  Section 3 "Cubic-graph edge contraction" (just the contraction
  definition + 4-face theorem).
- Restructure Section 4 to host both the original face-monochromatic-pair
  conjecture (clauses 1-3) and its strengthening (adds clause 4) as
  separate conjectures, after briefly experimenting with folding them
  into one. The empirical evidence is asymmetric (n<=21 for (1)-(3),
  n<=18 for the full set), which the two-conjecture split presents more
  honestly. The companion-paper reference is now in Section 4's intro.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-24 14:07:08 -04:00
parent 440ec9cc86
commit 83e9dba8ac
13 changed files with 759 additions and 427 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/dual_decomposition_minimal_counterexamples/experiments/draw_iterated_reduction.py → papers/dual_decomposition_iterated_reduction/experiments/draw_iterated_reduction.py
View File
papers/dual_decomposition_minimal_counterexamples/experiments/draw_iterated_reduction_n14.py → papers/dual_decomposition_iterated_reduction/experiments/draw_iterated_reduction_n14.py
View File
papers/dual_decomposition_minimal_counterexamples/fig_alg_step0.png → papers/dual_decomposition_iterated_reduction/fig_alg_step0.png
View File
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 77 KiB

After

Width:  |  Height:  |  Size: 77 KiB

papers/dual_decomposition_minimal_counterexamples/fig_alg_step1.png → papers/dual_decomposition_iterated_reduction/fig_alg_step1.png
View File
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 79 KiB

After

Width:  |  Height:  |  Size: 79 KiB

papers/dual_decomposition_minimal_counterexamples/fig_alg_step2.png → papers/dual_decomposition_iterated_reduction/fig_alg_step2.png
View File
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 77 KiB

After

Width:  |  Height:  |  Size: 77 KiB

papers/dual_decomposition_iterated_reduction/paper.aux
+34
View File
@@ -0,0 +1,34 @@
\relax
\citation{parent}
\citation{parent}
\citation{parent}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Setup and background}}{1}{}\protected@file@percent }
\newlabel{sec:background}{{1}{1}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The iterated reduction}}{1}{}\protected@file@percent }
\newlabel{sec:iterated-reduction}{{2}{1}}
\newlabel{alg:iterated-reduction}{{2.1}{1}}
\citation{parent}
\citation{parent}
\citation{parent}
\citation{parent}
\citation{parent}
\newlabel{rem:alg-invariants}{{2.2}{2}}
\newlabel{rem:alg-chord-apex}{{2.3}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Structural lemmas on the algorithm's output}}{2}{}\protected@file@percent }
\newlabel{sec:structural-lemmas}{{3}{2}}
\newlabel{lem:exactly-one-match}{{3.1}{2}}
\citation{parent}
\citation{parent}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Algorithm\nonbreakingspace 2.1\hbox {} on $G'=\mathrm {dual}(G)$, where $G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and $\varphi _1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies both the chord-apex and Kempe-cycle conditions of\nonbreakingspace \cite {parent}, found by \texttt {experiments/search\_kempe\_property.py}. \emph {Left:} $G'$ ($24$ vertices, $36$ edges) with the chosen pentagonal face shaded. \emph {Centre:} $H_1$ ($20$ vertices, $30$ edges) after step\nonbreakingspace (1) with $i_1 = 1$, $3$-edge-coloured by $\varphi _1$; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share the colour green. \emph {Right:} $H_2$ ($16$ vertices, $24$ edges) after step\nonbreakingspace (3) with $i_t = 3$; eight edges are protected, and the algorithm terminates one step later (no remaining safe pentagonal face in $H_2$). The generating script is \texttt {experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte barycentric embeddings with the outer face picked to keep $v_n^{(1)}, v_n^{(2)}$ in the interior.}}{3}{}\protected@file@percent }
\newlabel{fig:iterated-reduction-trace}{{1}{3}}
\newlabel{lem:all-distinct-exists}{{3.2}{3}}
\citation{parent}
\citation{parent}
\bibcite{parent}{1}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{12.7778pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
\gdef \@abspage@last{4}
papers/dual_decomposition_iterated_reduction/paper.log
+247
View File
@@ -0,0 +1,247 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 13:51
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
**paper.tex
(./paper.tex
LaTeX2e <2021-11-15> patch level 1
L3 programming layer <2022-02-24>
(/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls
Document Class: amsart 2020/05/29 v2.20.6
\linespacing=\dimen138
\normalparindent=\dimen139
\normaltopskip=\skip47
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
Package: amsmath 2021/10/15 v2.17l AMS math features
\@mathmargin=\skip48
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@=\dimen140
))
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
Package: amsbsy 1999/11/29 v1.2d Bold Symbols
\pmbraise@=\dimen141
)
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
Package: amsopn 2021/08/26 v2.02 operator names
)
\inf@bad=\count185
LaTeX Info: Redefining \frac on input line 234.
\uproot@=\count186
\leftroot@=\count187
LaTeX Info: Redefining \overline on input line 399.
\classnum@=\count188
\DOTSCASE@=\count189
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=\dimen142
LaTeX Font Info: Redeclaring font encoding OML on input line 743.
LaTeX Font Info: Redeclaring font encoding OMS on input line 744.
\macc@depth=\count190
\c@MaxMatrixCols=\count191
\dotsspace@=\muskip16
\c@parentequation=\count192
\dspbrk@lvl=\count193
\tag@help=\toks17
\row@=\count194
\column@=\count195
\maxfields@=\count196
\andhelp@=\toks18
\eqnshift@=\dimen143
\alignsep@=\dimen144
\tagshift@=\dimen145
\tagwidth@=\dimen146
\totwidth@=\dimen147
\lineht@=\dimen148
\@envbody=\toks19
\multlinegap=\skip49
\multlinetaggap=\skip50
\mathdisplay@stack=\toks20
LaTeX Info: Redefining \[ on input line 2938.
LaTeX Info: Redefining \] on input line 2939.
)
LaTeX Font Info: Trying to load font information for U+msa on input line 397
.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
)
(/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.
)
\copyins=\insert199
\abstractbox=\box52
\listisep=\skip51
\c@part=\count197
\c@section=\count198
\c@subsection=\count266
\c@subsubsection=\count267
\c@paragraph=\count268
\c@subparagraph=\count269
\c@figure=\count270
\c@table=\count271
\abovecaptionskip=\skip52
\belowcaptionskip=\skip53
\captionindent=\dimen149
\thm@style=\toks21
\thm@bodyfont=\toks22
\thm@headfont=\toks23
\thm@notefont=\toks24
\thm@headpunct=\toks25
\thm@preskip=\skip54
\thm@postskip=\skip55
\thm@headsep=\skip56
\dth@everypar=\toks26
)
(/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/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=\dimen150
\Gin@req@width=\dimen151
)
\c@theorem=\count272
(/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=\count273
\l__pdf_internal_box=\box53
)
(./paper.aux)
\openout1 = `paper.aux'.
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Trying to load font information for U+msa on input line 27.
(/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 27.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
)
(/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
\scratchcounter=\count274
\scratchdimen=\dimen152
\scratchbox=\box54
\nofMPsegments=\count275
\nofMParguments=\count276
\everyMPshowfont=\toks28
\MPscratchCnt=\count277
\MPscratchDim=\dimen153
\MPnumerator=\count278
\makeMPintoPDFobject=\count279
\everyMPtoPDFconversion=\toks29
) (/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
))
[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
<fig_alg_step0.png, id=18, 399.6106pt x 459.55217pt>
File: fig_alg_step0.png Graphic file (type png)
<use fig_alg_step0.png>
Package pdftex.def Info: fig_alg_step0.png used on input line 170.
(pdftex.def) Requested size: 115.20264pt x 132.48134pt.
<fig_alg_step1.png, id=20, 399.6106pt x 459.55217pt>
File: fig_alg_step1.png Graphic file (type png)
<use fig_alg_step1.png>
Package pdftex.def Info: fig_alg_step1.png used on input line 171.
(pdftex.def) Requested size: 115.20264pt x 132.48134pt.
<fig_alg_step2.png, id=21, 399.6106pt x 459.55217pt>
File: fig_alg_step2.png Graphic file (type png)
<use fig_alg_step2.png>
Package pdftex.def Info: fig_alg_step2.png used on input line 172.
(pdftex.def) Requested size: 115.20264pt x 132.48134pt.
LaTeX Warning: `h' float specifier changed to `ht'.
[2] [3 <./fig_alg_step0.png> <./fig_alg_step1.png> <./fig_alg_step2.png>]
[4] (./paper.aux) )
Here is how much of TeX's memory you used:
3022 strings out of 478268
42273 string characters out of 5846347
343206 words of memory out of 5000000
21067 multiletter control sequences out of 15000+600000
475834 words of font info for 54 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191
69i,7n,76p,1399b,294s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
nts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
s/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
m/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/
cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm
r5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.
pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/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/amsfonts/cm/cmsy5.pfb></
usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></us
r/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr
/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/l
ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/lo
cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (4 pages, 387470 bytes).
PDF statistics:
109 PDF objects out of 1000 (max. 8388607)
62 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000)
16 words of extra memory for PDF output out of 10000 (max. 10000000)
papers/dual_decomposition_iterated_reduction/paper.pdf
BIN
View File
Binary file not shown.
papers/dual_decomposition_iterated_reduction/paper.tex
+308
View File
@@ -0,0 +1,308 @@
%% filename: amsart-template.tex
%% American Mathematical Society
%% AMS-LaTeX v.2 template for use with amsart
%% ====================================================================
\documentclass{amsart}
\usepackage{amssymb}
\usepackage{graphicx}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{algorithm}[theorem]{Algorithm}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\begin{document}
\title{Iterated Reduction of Dual Minimal Counterexamples}
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
\thanks{}
\subjclass[2010]{Primary }
\keywords{four colour theorem, plane triangulation, dual graph, cubic planar
graph, edge connectivity, cyclic edge cut, Tait colouring,
$3$-edge-colouring}
\date{}
\dedicatory{}
\begin{abstract}
% TODO: abstract.
\end{abstract}
\maketitle
\section{Setup and background}
\label{sec:background}
This paper is a follow-up to \emph{Dual Decomposition of Minimal
Counterexamples}~\cite{parent}, which introduced the reduced-dual
construction: given a minimal counterexample $G$ to the Four Colour
Theorem, a degree-$5$ vertex $v$ of $G$ (equivalently a pentagonal face
$F_v$ of $G' = \mathrm{dual}(G)$), and an index $i \in \{0,1,2,3,4\}$, the
\emph{reduced dual} $\widehat{G}'_{v,i}$ is the cubic plane graph obtained
from $G'$ by deleting the five boundary vertices of $F_v$, listing the
resulting five degree-$2$ vertices clockwise as $A_0,\dots,A_4$ along the
new face $F$, attaching a new apex vertex $v_n$ to $A_i, A_{i+1}, A_{i+2}$
by three new edges, and adding the chord $A_{i+3} A_{i+4}$. The four edges
added by steps~(3) and~(4) are named the \emph{side-$0$ edge}
($v_n A_i$), the \emph{spike edge} ($v_n A_{i+1}$), the \emph{side-$1$
edge} ($v_n A_{i+2}$), and the \emph{merged edge} ($A_{i+3} A_{i+4}$). The
parent paper also proves two structural lemmas about every proper
$3$-edge-colouring $\varphi$ of $\widehat{G}'_{v,i}$:
\begin{itemize}
\item the \emph{chord-apex lemma}, asserting
$\varphi(\mathrm{spike}) = \varphi(\mathrm{merged})$;
\item the \emph{Kempe-cycle lemma}, asserting that
the spike and merged edge lie on a common
$\{\varphi(\mathrm{spike}), \varphi(\mathrm{side-}j)\}$-Kempe
cycle through the side-$j$ edge for both $j = 0, 1$.
\end{itemize}
We refer the reader to~\cite{parent} for the precise definitions, proofs,
and the pentagonal-externals lemma we will reuse below.
\section{The iterated reduction}
\label{sec:iterated-reduction}
The reduced-dual construction can be iterated: starting from a proper
$3$-edge-colouring $\varphi_1$ of a reduced dual $\widehat{G}'_{v,i}$, we
apply the construction again to that graph at a pentagonal face whose ten
incident edges avoid the four named edges from the first reduction,
extending $\varphi_1$ across the new reduction. The protected edges
accumulate into a set $E$ that grows by four per iteration, and the
process terminates when $E$ has blocked every pentagonal face.
\begin{algorithm}[Iterated reduction with protected edges]
\label{alg:iterated-reduction}
Let $G$ be a triangulation we assume to be a minimal counterexample to the
Four Colour Theorem. The algorithm produces a sequence $H_1, H_2, \dots$ of
cubic plane graphs, proper $3$-edge-colourings $\varphi_t$ of $H_t$, and a
growing set $E$ of protected edges.
\begin{enumerate}
\item[(0)] Form $G' := \mathrm{dual}(G)$, a cubic plane graph.
\item[(1)] Choose a degree-$5$ vertex $v$ of $G$ (equivalently a
pentagonal face $F_v$ of $G'$) and an index
$i_1 \in \{0, \dots, 4\}$. Apply the reduced-dual construction
of~\cite{parent} to form $H_1 := \widehat{G'}_{v, i_1}$, and fix
any proper $3$-edge-colouring $\varphi_1$ of $H_1$ (one exists
by the minimality of $G$).
\item[(2)] Initialise
$E := \{\text{spike}, \text{side-}0, \text{side-}1,
\text{merged}\}$, the four named edges of the reduction in~(1).
\item[(3)] (Iterate.) At step $t \geq 2$, given $H_{t-1}$,
$\varphi_{t-1}$, and $E \subseteq E(H_{t-1})$:
\begin{enumerate}
\item[(a)] Find a pentagonal face $F$ of $H_{t-1}$ whose ten
incident edges --- the five boundary edges of $\partial F$
and the five external edges at $\partial F$ --- are all
outside $E$. If no such $F$ exists, terminate.
\item[(b)] By the pentagonal-externals lemma of~\cite{parent}
applied to $H_{t-1}$ at $F$ under $\varphi_{t-1}$, the
external vector has shape $(a, b, c, c, c)$ up to cyclic
rotation. Choose an index $i_t$ for which
$\varphi_{t-1}(f_{i_t + 3}) = \varphi_{t-1}(f_{i_t + 4})$
and $\varphi_{t-1}(f_{i_t})$, $\varphi_{t-1}(f_{i_t + 1})$,
$\varphi_{t-1}(f_{i_t + 2})$ are three distinct colours.
\item[(c)] Apply the reduced-dual construction
of~\cite{parent} to $H_{t-1}$ at $(F, i_t)$ to form $H_t$.
\item[(d)] Extend $\varphi_{t-1}$ to a proper
$3$-edge-colouring $\varphi_t$ of $H_t$: every surviving
edge keeps its $\varphi_{t-1}$-colour, and each new edge
takes the unique colour completing the palette at its
endpoint (consistent across both endpoints of the chord
by the choice of $i_t$).
\item[(e)] Add the four named edges of the step-$t$ reduction
to $E$.
\end{enumerate}
\item[(4)] Repeat (3) until termination.
\end{enumerate}
\end{algorithm}
\begin{remark}
\label{rem:alg-invariants}
At each iteration, $|V(H_t)| = |V(H_{t-1})| - 4$ and
$|E(H_t)| = |E(H_{t-1})| - 6$, so $H_t$ shrinks at a fixed rate; the
protected set $|E|$ grows by exactly four; and every protected edge
survives all subsequent reductions. Since the graph is finite,
termination is guaranteed. By the pentagonal-externals lemma
of~\cite{parent}, step~(b) never fails: some valid $i_t$ always exists
for any pentagonal face under any proper colouring. Termination is
therefore combinatorial: it occurs precisely when $E$ touches every
pentagonal face of $H_{t-1}$.
\end{remark}
\begin{remark}
\label{rem:alg-chord-apex}
The chord-apex lemma of~\cite{parent} applies only at $t = 1$, when $H_1$
is a reduced dual of $G'$. For $t \geq 2$, $H_t$ is a reduced dual of
$H_{t-1}$ rather than of $G'$, and $H_{t-1}$ is itself
$3$-edge-colourable, so the non-$3$-edge-colourability argument that
drives the chord-apex lemma does not carry over. Whether the constraints
accumulated in $E$ propagate any further structure to $\varphi_t$ for
$t \geq 2$ is left open.
\end{remark}
\begin{figure}[h]
\centering
\includegraphics[width=0.32\textwidth]{fig_alg_step0.png}\hfill
\includegraphics[width=0.32\textwidth]{fig_alg_step1.png}\hfill
\includegraphics[width=0.32\textwidth]{fig_alg_step2.png}
\caption{Algorithm~\ref{alg:iterated-reduction} on
$G'=\mathrm{dual}(G)$, where $G$ is the first min-degree-$5$ plantri
triangulation on $14$ vertices and $\varphi_1$ is a specific proper
$3$-edge-colouring of $H_1$ that satisfies both the chord-apex and
Kempe-cycle conditions of~\cite{parent}, found by
\texttt{experiments/search\_kempe\_property.py}. \emph{Left:} $G'$
($24$ vertices, $36$ edges) with the chosen pentagonal face shaded.
\emph{Centre:} $H_1$ ($20$ vertices, $30$ edges) after step~(1) with
$i_1 = 1$, $3$-edge-coloured by $\varphi_1$; the four edges around
$v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges
share the colour green. \emph{Right:} $H_2$ ($16$ vertices, $24$ edges)
after step~(3) with $i_t = 3$; eight edges are protected, and the
algorithm terminates one step later (no remaining safe pentagonal face
in $H_2$). The generating script is
\texttt{experiments/draw\_iterated\_reduction\_n14.py}; layouts are
Tutte barycentric embeddings with the outer face picked to keep
$v_n^{(1)}, v_n^{(2)}$ in the interior.}
\label{fig:iterated-reduction-trace}
\end{figure}
\section{Structural lemmas on the algorithm's output}
\label{sec:structural-lemmas}
\begin{lemma}[Exactly one matching pair in the algorithm's output]
\label{lem:exactly-one-match}
Let $G$ be a minimal counterexample to the Four Colour Theorem, and let
$(H_{t^*}, \varphi_{t^*})$ be the final graph-and-colouring produced by
some terminating execution of
Algorithm~\ref{alg:iterated-reduction} on $G$, with named pairs
$(\mathrm{spike}_t, \mathrm{merged}_t)$ for $t = 1, \dots, t^*$. Then
there is exactly one $t$ with
$\varphi_{t^*}(\mathrm{spike}_t) = \varphi_{t^*}(\mathrm{merged}_t)$, and
it is $t = 1$.
\end{lemma}
\begin{proof}
The algorithm never re-colours an existing edge: at each iteration
step~(3d) every surviving edge keeps its $\varphi_{t-1}$-colour, and the
four new edges receive fresh colours forced by propriety. Hence for every
$1 \leq k \leq t \leq t^*$,
\[
\varphi_t(\mathrm{spike}_k) = \varphi_k(\mathrm{spike}_k),
\qquad
\varphi_t(\mathrm{merged}_k) = \varphi_k(\mathrm{merged}_k);
\]
the colours of the step-$k$ named edges, once written, are permanent. It
suffices to compare $\varphi_k(\mathrm{spike}_k)$ and
$\varphi_k(\mathrm{merged}_k)$ at the step where each pair is introduced.
\textbf{Case $k = 1$.} Since $G$ is a minimal counterexample, $H_1$ is a
reduced dual of $G'$. The chord-apex lemma of~\cite{parent} applied to
$\varphi_1$ gives
$\varphi_1(\mathrm{spike}_1) = \varphi_1(\mathrm{merged}_1)$.
\textbf{Case $k \geq 2$.} At step $k$ the algorithm picks an index $i_k$
for which $f_{i_k+3} = f_{i_k+4}$ (chord consistency) and
$f_{i_k}, f_{i_k+1}, f_{i_k+2}$ are pairwise distinct (propriety at the
new $v_n$), where $f$ is the external vector of the chosen pentagonal
face of $H_{k-1}$ under $\varphi_{k-1}$. Step~(3d) then assigns
\[
\varphi_k(\mathrm{spike}_k) = f_{i_k+1},
\qquad
\varphi_k(\mathrm{merged}_k) = f_{i_k+3}.
\]
By the pentagonal-externals lemma of~\cite{parent}, $f$ has the $(2,2,1)$
pattern: a block of three consecutive positions $\{p, p+1, p+2\}$
(mod $5$) on which it is constantly some colour $c$, while the remaining
two positions $\{p+3, p+4\}$ hold the two non-$c$ colours, one each. The
condition $f_{i_k+3} = f_{i_k+4}$ forces $(i_k+3, i_k+4)$ to be either
$(p, p+1)$ or $(p+1, p+2)$ --- the two consecutive pairs inside the
block --- and correspondingly $i_k + 1 \in \{p+3, p+4\}$,
\emph{outside} the block. So $f_{i_k+1}$ is not $c$, whereas
$f_{i_k+3} = c$, and hence
$\varphi_k(\mathrm{spike}_k) \neq \varphi_k(\mathrm{merged}_k)$.
Combining the two cases, exactly one $t \in \{1, \dots, t^*\}$ --- namely
$t = 1$ --- has $\varphi_{t^*}(\mathrm{spike}_t) =
\varphi_{t^*}(\mathrm{merged}_t)$.
\end{proof}
\begin{lemma}[All-distinct colouring exists on a 4-colourable graph]
\label{lem:all-distinct-exists}
Let $G$ be a $4$-colourable maximal planar graph of minimum degree
$\geq 5$ (equivalently, a maximal planar graph that is \emph{not} a
minimal counterexample to the Four Colour Theorem). Then there is an
execution of Algorithm~\ref{alg:iterated-reduction} on $G$ whose final
colouring $\varphi_{t^*}$ satisfies
$\varphi_{t^*}(\mathrm{spike}_t) \neq \varphi_{t^*}(\mathrm{merged}_t)$
for every $t \in \{1, \dots, t^*\}$. In particular, there exists a proper
$3$-edge-colouring of $H_{t^*}$ under which every spike-merged pair has
distinct colours.
\end{lemma}
\begin{proof}
The argument mirrors Lemma~\ref{lem:exactly-one-match}, but extends a
colouring \emph{downward} from $G'$ rather than carrying one forward from
$H_1$.
Since $G$ is $4$-colourable, by Tait's theorem
$G' = \mathrm{dual}(G)$ admits a proper $3$-edge-colouring $\xi$. Apply
the pentagonal-externals lemma of~\cite{parent} to $\xi$ at the
pentagonal face $F_v$ selected in step~(1): the external vector
$f = (f_0, \dots, f_4)$ at $F_v$ under $\xi$ has the $(3,1,1)$
cyclic-consecutive shape, with a block of three consecutive positions
$\{p, p+1, p+2\}$ (mod $5$) holding a common colour $c$, and the
remaining two positions $\{p+3, p+4\}$ holding the two non-$c$ colours,
one each. The algorithm's choice of $i_1$ forces $\{i_1+3, i_1+4\}$
inside the $c$-block (so the chord is consistently coloured) and the
three positions $\{i_1, i_1+1, i_1+2\}$ pairwise distinct; in particular
$i_1+1$ lies \emph{outside} the $c$-block.
Choose $\varphi_1$ to be the proper $3$-edge-colouring of $H_1$ that
agrees with $\xi$ on every surviving edge and assigns each new edge at
$A_j$ the unique third colour at $A_j$. Then
$\varphi_1(\mathrm{spike}_1) = f_{i_1+1}$, a value not equal to $c$,
while $\varphi_1(\mathrm{merged}_1) = f_{i_1+3} = c$, so
$\varphi_1(\mathrm{spike}_1) \neq \varphi_1(\mathrm{merged}_1)$.
The same argument repeats at every step $k \geq 2$: the external vector
at the chosen pentagonal face under $\varphi_{k-1}$ has the $(3,1,1)$
cyclic-consecutive shape (pentagonal-externals lemma of~\cite{parent}),
the algorithm's index choice $i_k$ puts $i_k+3, i_k+4$ inside the colour
block and $i_k+1$ outside, and step~(3d) thus assigns
$\varphi_k(\mathrm{spike}_k) \neq \varphi_k(\mathrm{merged}_k)$. The
algorithm preserves these colours through every later step, so
$\varphi_{t^*}(\mathrm{spike}_t) \neq \varphi_{t^*}(\mathrm{merged}_t)$
for every $t \in \{1, \dots, t^*\}$.
\end{proof}
\begin{thebibliography}{9}
\bibitem{parent}
E.~Bauerfeld, \emph{Dual Decomposition of Minimal Counterexamples}.
Companion paper.
\end{thebibliography}
\end{document}
papers/dual_decomposition_minimal_counterexamples/paper.aux
+15 -22
View File
@@ -13,30 +13,23 @@
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 2.6\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 2.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{5}{}\protected@file@percent }
\newlabel{fig:chord-apex-proof}{{2}{5}}
\newlabel{lem:kempe-spike}{{2.7}{6}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{An iterated reduction}}{7}{}\protected@file@percent }
\newlabel{alg:iterated-reduction}{{3.1}{7}}
\newlabel{rem:alg-invariants}{{3.2}{7}}
\newlabel{rem:alg-chord-apex}{{3.3}{7}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Algorithm\nonbreakingspace 3.1\hbox {} on $G'=\mathrm {dual}(G)$, where $G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and $\varphi _1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies both the chord-apex condition (Lemma\nonbreakingspace 2.6\hbox {}) and the Kempe-cycle condition (Lemma\nonbreakingspace 2.7\hbox {}), found by \texttt {experiments/search\_kempe\_property.py}. \emph {Left:} $G'$ ($24$ vertices, $36$ edges) with the chosen pentagonal face shaded. \emph {Centre:} $H_1$ ($20$ vertices, $30$ edges) after step\nonbreakingspace (1) with $i_1 = 1$, $3$-edge-coloured by $\varphi _1$; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share the colour green. \emph {Right:} $H_2$ ($16$ vertices, $24$ edges) after step\nonbreakingspace (3) with $i_t = 3$; eight edges are protected, and the algorithm terminates one step later (no remaining safe pentagonal face in $H_2$). The generating script is \texttt {experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte barycentric embeddings with the outer face picked to keep $v_n^{(1)}, v_n^{(2)}$ in the interior.}}{8}{}\protected@file@percent }
\newlabel{fig:iterated-reduction-trace}{{3}{8}}
\newlabel{lem:exactly-one-match}{{3.4}{8}}
\newlabel{lem:all-distinct-exists}{{3.5}{9}}
\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{3.6}{9}}
\newlabel{rem:conj-3-6-empirical}{{3.7}{10}}
\newlabel{def:cubic-edge-contraction}{{3.8}{10}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Cubic-graph edge contraction (Definition\nonbreakingspace 3.8\hbox {}). Left: a fragment of a cubic plane graph with the contracted edge $e = uv$ highlighted in red. Middle: deleting $e$ leaves $u$ and $v$ of degree\nonbreakingspace $2$. Right: smoothing $u$ and $v$ replaces each pair of incident edges by a single new edge, removing $u, v$ and giving a cubic plane graph again.}}{11}{}\protected@file@percent }
\newlabel{fig:cubic-edge-contraction}{{4}{11}}
\newlabel{thm:cubic-contraction-4face}{{3.9}{11}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 3.9\hbox {}. Left: the $4$-face $f$ of $H$ under $\varphi $, with the forced colours $\varphi (e_0) = a$, $\varphi (e_1) = b$, $\varphi (e_2) = \varphi (e_3) = c$, $\varphi (w_0) = \varphi (w_1) = b$, and $\varphi (w_2) = \varphi (w_3) = a$. Right: the contracted graph $H'$ under $\varphi '$. The smoothed-in edges $e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is recoloured from $b$ to $c$; every edge outside the face neighbourhood keeps its $\varphi $-colour (dotted in red: the five edges of $H$ removed by the contraction).}}{12}{}\protected@file@percent }
\newlabel{fig:thm-cubic-contraction-4face}{{5}{12}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The Four Colour Theorem from a strengthened conjecture}}{12}{}\protected@file@percent }
\newlabel{sec:toward-4ct}{{4}{12}}
\newlabel{conj:face-monochromatic-pair-strengthened}{{4.1}{12}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Cubic-graph edge contraction}}{7}{}\protected@file@percent }
\newlabel{def:cubic-edge-contraction}{{3.1}{7}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Cubic-graph edge contraction (Definition\nonbreakingspace 3.1\hbox {}). Left: a fragment of a cubic plane graph with the contracted edge $e = uv$ highlighted in red. Middle: deleting $e$ leaves $u$ and $v$ of degree\nonbreakingspace $2$. Right: smoothing $u$ and $v$ replaces each pair of incident edges by a single new edge, removing $u, v$ and giving a cubic plane graph again.}}{7}{}\protected@file@percent }
\newlabel{fig:cubic-edge-contraction}{{3}{7}}
\newlabel{thm:cubic-contraction-4face}{{3.2}{7}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{8}{}\protected@file@percent }
\newlabel{sec:toward-4ct}{{4}{8}}
\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{4.1}{8}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 3.2\hbox {}. Left: the $4$-face $f$ of $H$ under $\varphi $, with the forced colours $\varphi (e_0) = a$, $\varphi (e_1) = b$, $\varphi (e_2) = \varphi (e_3) = c$, $\varphi (w_0) = \varphi (w_1) = b$, and $\varphi (w_2) = \varphi (w_3) = a$. Right: the contracted graph $H'$ under $\varphi '$. The smoothed-in edges $e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is recoloured from $b$ to $c$; every edge outside the face neighbourhood keeps its $\varphi $-colour (dotted in red: the five edges of $H$ removed by the contraction).}}{9}{}\protected@file@percent }
\newlabel{fig:thm-cubic-contraction-4face}{{4}{9}}
\newlabel{rem:conj-3-6-empirical}{{4.2}{9}}
\newlabel{conj:face-monochromatic-pair-strengthened}{{4.3}{10}}
\newlabel{rem:conj-3-8-empirical}{{4.4}{10}}
\newlabel{rem:implication-4ct}{{4.5}{10}}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{0pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\newlabel{rem:conj-3-8-empirical}{{4.2}{13}}
\newlabel{rem:implication-4ct}{{4.3}{13}}
\gdef \@abspage@last{14}
\gdef \@abspage@last{11}
papers/dual_decomposition_minimal_counterexamples/paper.log
+41 -91
View File
@@ -1,4 +1,4 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 13:42
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 14:04
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
@@ -244,119 +244,69 @@ LaTeX Warning: `h' float specifier changed to `ht'.
[4] [5 <./fig_chord_apex_step1.png> <./fig_chord_apex_step2.png> <./fig_chord_a
pex_step3.png>] [6]
Overfull \hbox (4.76643pt too wide) in paragraph at lines 433--440
\OT1/cmr/m/n/10 which $\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[
]\OT1/cmr/m/n/10 ) = \OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\
OT1/cmr/m/n/10 )$ and $\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[
]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 ; '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\O
T1/cmr/m/n/10 )\OML/cmm/m/it/10 ; '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\OT1/
cmr/m/n/10 )$
[]
<fig_alg_step0.png, id=49, 399.6106pt x 459.55217pt>
File: fig_alg_step0.png Graphic file (type png)
<use fig_alg_step0.png>
Package pdftex.def Info: fig_alg_step0.png used on input line 479.
(pdftex.def) Requested size: 115.20264pt x 132.48134pt.
<fig_alg_step1.png, id=50, 399.6106pt x 459.55217pt>
File: fig_alg_step1.png Graphic file (type png)
<use fig_alg_step1.png>
Package pdftex.def Info: fig_alg_step1.png used on input line 480.
(pdftex.def) Requested size: 115.20264pt x 132.48134pt.
<fig_alg_step2.png, id=51, 399.6106pt x 459.55217pt>
File: fig_alg_step2.png Graphic file (type png)
<use fig_alg_step2.png>
Package pdftex.def Info: fig_alg_step2.png used on input line 481.
(pdftex.def) Requested size: 115.20264pt x 132.48134pt.
Underfull \hbox (badness 4391) in paragraph at lines 498--498
\OT1/cmr/m/sc/10 Figure 3.\OT1/cmr/m/n/10 Algorithm 3.1[] on $\OML/cmm/m/it/10
G[] \OT1/cmr/m/n/10 = [](\OML/cmm/m/it/10 G\OT1/cmr/m/n/10 )$, where $\OML/cmm/
m/it/10 G$
[]
Underfull \hbox (badness 3623) in paragraph at lines 498--498
\OT1/cmr/m/n/10 is the first min-degree-$5$ plantri tri-an-gu-la-tion on $14$ v
er-
[]
Underfull \hbox (badness 3179) in paragraph at lines 498--498
\OT1/cmr/m/n/10 tices and $\OML/cmm/m/it/10 '[]$ \OT1/cmr/m/n/10 is a spe-cific
proper $3$-edge-colouring of $\OML/cmm/m/it/10 H[]$
[]
Underfull \hbox (badness 3209) in paragraph at lines 498--498
\OT1/cmr/m/n/10 that sat-is-fies both the chord-apex con-di-tion (Lemma 2.6[])
[]
Underfull \hbox (badness 6094) in paragraph at lines 498--498
\OT1/cmr/m/n/10 and the Kempe-cycle con-di-tion (Lemma 2.7[]), found by
[]
[7] [8 <./fig_alg_step0.png> <./fig_alg_step1.png> <./fig_alg_step2.png>]
[9] [10]
<fig_cubic_edge_contraction.png, id=75, 950.752pt x 203.159pt>
<fig_cubic_edge_contraction.png, id=49, 950.752pt x 203.159pt>
File: fig_cubic_edge_contraction.png Graphic file (type png)
<use fig_cubic_edge_contraction.png>
Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 702
Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 432
.
(pdftex.def) Requested size: 341.9989pt x 73.08138pt.
[11 <./fig_cubic_edge_contraction.png>]
<fig_thm_cubic_contraction_4face.png, id=80, 916.223pt x 417.56pt>
[7 <./fig_cubic_edge_contraction.png>]
<fig_thm_cubic_contraction_4face.png, id=55, 916.223pt x 417.56pt>
File: fig_thm_cubic_contraction_4face.png Graphic file (type png)
<use fig_thm_cubic_contraction_4face.png>
Package pdftex.def Info: fig_thm_cubic_contraction_4face.png used on input lin
e 777.
e 507.
(pdftex.def) Requested size: 352.79846pt x 160.78339pt.
[12 <./fig_thm_cubic_contraction_4face.png>]
Underfull \hbox (badness 1648) in paragraph at lines 837--842
\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 4.2\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T
LaTeX Warning: `h' float specifier changed to `ht'.
[8]
Underfull \vbox (badness 10000) has occurred while \output is active []
[9 <./fig_thm_cubic_contraction_4face.png>]
Underfull \hbox (badness 1648) in paragraph at lines 638--644
\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 4.4\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T
he strength-ened con-jec-ture was tested on the same chord-
[]
Underfull \hbox (badness 1014) in paragraph at lines 837--842
\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 3.7[]; for each colour-ing we
Underfull \hbox (badness 1014) in paragraph at lines 638--644
\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 4.2[]; for each colour-ing we
sought any
[]
[13] [14] (./paper.aux) )
[10] [11] (./paper.aux) )
Here is how much of TeX's memory you used:
3114 strings out of 478268
44776 string characters out of 5846347
346374 words of memory out of 5000000
21145 multiletter control sequences out of 15000+600000
3087 strings out of 478268
44184 string characters out of 5846347
350310 words of memory out of 5000000
21121 multiletter control sequences out of 15000+600000
478077 words of font info for 62 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191
69i,9n,76p,1306b,326s stack positions out of 10000i,1000n,20000p,200000b,200000s
69i,9n,76p,744b,373s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf-dist/fonts/type1/public
/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/
amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
msfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
sfonts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
onts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmmi9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
m/cmr5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c
mr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8
.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr9.pf
b></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb><
/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></u
sr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb></usr
/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/
local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/lo
cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/loc
al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (14 pages, 1148116 bytes).
sfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
onts/cm/cmmi9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm
/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cm
r9.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy1
0.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.
pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pf
b></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb
></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb><
/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></
usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pf
b>
Output written on paper.pdf (11 pages, 941151 bytes).
PDF statistics:
175 PDF objects out of 1000 (max. 8388607)
94 compressed objects within 1 object stream
144 PDF objects out of 1000 (max. 8388607)
78 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000)
61 words of extra memory for PDF output out of 10000 (max. 10000000)
46 words of extra memory for PDF output out of 10000 (max. 10000000)
papers/dual_decomposition_minimal_counterexamples/paper.pdf
BIN
View File
Binary file not shown.
papers/dual_decomposition_minimal_counterexamples/paper.tex
+114 -314
View File
@@ -393,281 +393,11 @@ distinct colours, contradicting Lemma~\ref{lem:chord-apex} applied to
$\varphi'$.
\end{proof}
\section{An iterated reduction}
\section{Cubic-graph edge contraction}
The reduced-dual construction in Definition~\ref{def:reduced-dual} can be
iterated: starting from a proper $3$-edge-colouring $\varphi_1$ of a reduced
dual $\widehat{G}'_{v,i}$, we apply the construction again to that graph at
a pentagonal face whose ten incident edges avoid the four named edges from
the first reduction, extending $\varphi_1$ across the new reduction. The
protected edges accumulate into a set $E$ that grows by four per iteration,
and the process terminates when $E$ has blocked every pentagonal face.
\begin{algorithm}[Iterated reduction with protected edges]
\label{alg:iterated-reduction}
Let $G$ be a triangulation we assume to be a minimal counterexample to the
Four Colour Theorem. The algorithm produces a sequence $H_1, H_2, \dots$ of
cubic plane graphs, proper $3$-edge-colourings $\varphi_t$ of $H_t$, and a
growing set $E$ of protected edges.
\begin{enumerate}
\item[(0)] Form $G' := \mathrm{dual}(G)$, a cubic plane graph.
\item[(1)] Choose a degree-$5$ vertex $v$ of $G$ (equivalently a pentagonal
face $F_v$ of $G'$) and an index $i_1 \in \{0, \dots, 4\}$. Apply
Definition~\ref{def:reduced-dual} to form
$H_1 := \widehat{G'}_{v, i_1}$, and fix any proper
$3$-edge-colouring $\varphi_1$ of $H_1$ (one exists by the minimality
of $G$).
\item[(2)] Initialise $E := \{\text{spike}, \text{side-}0, \text{side-}1,
\text{merged}\}$, the four named edges of the reduction in (1).
\item[(3)] (Iterate.) At step $t \geq 2$, given $H_{t-1}$, $\varphi_{t-1}$,
and $E \subseteq E(H_{t-1})$:
\begin{enumerate}
\item[(a)] Find a pentagonal face $F$ of $H_{t-1}$ whose ten
incident edges --- the five boundary edges of $\partial F$
and the five external edges at $\partial F$ --- are all
outside $E$. If no such $F$ exists, terminate.
\item[(b)] By Lemma~\ref{lem:pentagonal-externals} applied to
$H_{t-1}$ at $F$ under $\varphi_{t-1}$, the external vector
has shape $(a, b, c, c, c)$ up to cyclic rotation. Choose an
index $i_t$ for which
$\varphi_{t-1}(f_{i_t + 3}) = \varphi_{t-1}(f_{i_t + 4})$ and
$\varphi_{t-1}(f_{i_t}), \varphi_{t-1}(f_{i_t + 1}),
\varphi_{t-1}(f_{i_t + 2})$ are three distinct colours.
\item[(c)] Apply Definition~\ref{def:reduced-dual} to $H_{t-1}$ at
$(F, i_t)$ to form $H_t$.
\item[(d)] Extend $\varphi_{t-1}$ to a proper $3$-edge-colouring
$\varphi_t$ of $H_t$: every surviving edge keeps its
$\varphi_{t-1}$-colour, and each new edge takes the unique
colour completing the palette at its endpoint (consistent
across both endpoints of the chord by the choice of $i_t$).
\item[(e)] Add the four named edges of the step-$t$ reduction to
$E$.
\end{enumerate}
\item[(4)] Repeat (3) until termination.
\end{enumerate}
\end{algorithm}
\begin{remark}
\label{rem:alg-invariants}
At each iteration, $|V(H_t)| = |V(H_{t-1})| - 4$ and
$|E(H_t)| = |E(H_{t-1})| - 6$, so $H_t$ shrinks at a fixed rate; the
protected set $|E|$ grows by exactly four; and every protected edge survives
all subsequent reductions. Since the graph is finite, termination is
guaranteed. By Lemma~\ref{lem:pentagonal-externals}, step~(b) never fails:
some valid $i_t$ always exists for any pentagonal face under any proper
colouring. Termination is therefore combinatorial: it occurs precisely when
$E$ touches every pentagonal face of $H_{t-1}$.
\end{remark}
\begin{remark}
\label{rem:alg-chord-apex}
Lemma~\ref{lem:chord-apex} applies only at $t = 1$, when $H_1$ is a reduced
dual of $G'$. For $t \geq 2$, $H_t$ is a reduced dual of $H_{t-1}$ rather than
of $G'$, and $H_{t-1}$ is itself $3$-edge-colourable, so the
non-$3$-edge-colourability argument that drives Lemma~\ref{lem:chord-apex}
does not carry over. Whether the constraints accumulated in $E$ propagate
any further structure to $\varphi_t$ for $t \geq 2$ is left open.
\end{remark}
\begin{figure}[h]
\centering
\includegraphics[width=0.32\textwidth]{fig_alg_step0.png}\hfill
\includegraphics[width=0.32\textwidth]{fig_alg_step1.png}\hfill
\includegraphics[width=0.32\textwidth]{fig_alg_step2.png}
\caption{Algorithm~\ref{alg:iterated-reduction} on $G'=\mathrm{dual}(G)$, where
$G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and
$\varphi_1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies
both the chord-apex condition (Lemma~\ref{lem:chord-apex}) and the Kempe-cycle
condition (Lemma~\ref{lem:kempe-spike}), found by
\texttt{experiments/search\_kempe\_property.py}. \emph{Left:} $G'$
($24$ vertices, $36$ edges) with the chosen pentagonal face shaded.
\emph{Centre:} $H_1$ ($20$ vertices, $30$ edges) after step~(1) with
$i_1 = 1$, $3$-edge-coloured by $\varphi_1$; the four edges around
$v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share
the colour green. \emph{Right:} $H_2$ ($16$ vertices, $24$ edges) after
step~(3) with $i_t = 3$; eight edges are protected, and the algorithm
terminates one step later (no remaining safe pentagonal face in $H_2$).
The generating script is
\texttt{experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte
barycentric embeddings with the outer face picked to keep $v_n^{(1)},
v_n^{(2)}$ in the interior.}
\label{fig:iterated-reduction-trace}
\end{figure}
\begin{lemma}[Exactly one matching pair in the algorithm's output]
\label{lem:exactly-one-match}
Let $G$ be a minimal counterexample to the Four Colour Theorem, and let
$(H_{t^*}, \varphi_{t^*})$ be the final graph-and-colouring produced by some
terminating execution of Algorithm~\ref{alg:iterated-reduction} on $G$, with
named pairs $(\mathrm{spike}_t, \mathrm{merged}_t)$ for $t = 1, \dots, t^*$.
Then there is exactly one $t$ with
$\varphi_{t^*}(\mathrm{spike}_t) = \varphi_{t^*}(\mathrm{merged}_t)$, and it
is $t = 1$.
\end{lemma}
\begin{proof}
The algorithm never re-colours an existing edge: at each iteration
step~(3d) every surviving edge keeps its $\varphi_{t-1}$-colour, and the
four new edges receive fresh colours forced by propriety. Hence for every
$1 \leq k \leq t \leq t^*$,
\[
\varphi_t(\mathrm{spike}_k) = \varphi_k(\mathrm{spike}_k),
\qquad
\varphi_t(\mathrm{merged}_k) = \varphi_k(\mathrm{merged}_k);
\]
the colours of the step-$k$ named edges, once written, are permanent. It
suffices to compare $\varphi_k(\mathrm{spike}_k)$ and
$\varphi_k(\mathrm{merged}_k)$ at the step where each pair is introduced.
\textbf{Case $k = 1$.} Since $G$ is a minimal counterexample, $H_1$ is a
reduced dual of $G'$. Lemma~\ref{lem:chord-apex} applied to $\varphi_1$ gives
$\varphi_1(\mathrm{spike}_1) = \varphi_1(\mathrm{merged}_1)$.
\textbf{Case $k \geq 2$.} At step $k$ the algorithm picks an index $i_k$ for
which $f_{i_k+3} = f_{i_k+4}$ (chord consistency) and $f_{i_k}, f_{i_k+1},
f_{i_k+2}$ are pairwise distinct (propriety at the new $v_n$), where $f$ is
the external vector of the chosen pentagonal face of $H_{k-1}$ under
$\varphi_{k-1}$. Step~(3d) then assigns
\[
\varphi_k(\mathrm{spike}_k) = f_{i_k+1},
\qquad
\varphi_k(\mathrm{merged}_k) = f_{i_k+3}.
\]
By Lemma~\ref{lem:pentagonal-externals}, $f$ has the $(2,2,1)$ pattern: a
block of three consecutive positions $\{p, p+1, p+2\}$ (mod $5$) on which it
is constantly some colour $c$, while the remaining two positions
$\{p+3, p+4\}$ hold the two non-$c$ colours, one each. The condition
$f_{i_k+3} = f_{i_k+4}$ forces $(i_k+3, i_k+4)$ to be either $(p, p+1)$ or
$(p+1, p+2)$ --- the two consecutive pairs inside the block --- and
correspondingly $i_k + 1 \in \{p+3, p+4\}$, \emph{outside} the block. So
$f_{i_k+1}$ is not $c$, whereas $f_{i_k+3} = c$, and hence
$\varphi_k(\mathrm{spike}_k) \neq \varphi_k(\mathrm{merged}_k)$.
Combining the two cases, exactly one $t \in \{1, \dots, t^*\}$ --- namely
$t = 1$ --- has $\varphi_{t^*}(\mathrm{spike}_t) =
\varphi_{t^*}(\mathrm{merged}_t)$.
\end{proof}
\begin{lemma}[All-distinct colouring exists on a 4-colourable graph]
\label{lem:all-distinct-exists}
Let $G$ be a $4$-colourable maximal planar graph of minimum degree $\geq 5$
(equivalently, a maximal planar graph that is \emph{not} a minimal
counterexample to the Four Colour Theorem). Then there is an execution of
Algorithm~\ref{alg:iterated-reduction} on $G$ whose final colouring
$\varphi_{t^*}$ satisfies
$\varphi_{t^*}(\mathrm{spike}_t) \neq \varphi_{t^*}(\mathrm{merged}_t)$ for
every $t \in \{1, \dots, t^*\}$. In particular, there exists a proper
$3$-edge-colouring of $H_{t^*}$ under which every spike-merged pair has
distinct colours.
\end{lemma}
\begin{proof}
The argument mirrors Lemma~\ref{lem:exactly-one-match}, but extends a
colouring \emph{downward} from $G'$ rather than carrying one forward from
$H_1$.
Since $G$ is $4$-colourable, by Tait's theorem
$G' = \mathrm{dual}(G)$ admits a proper $3$-edge-colouring $\xi$. Apply
Lemma~\ref{lem:pentagonal-externals} to $\xi$ at the pentagonal face $F_v$
selected in step~(1): the external vector $f = (f_0, \dots, f_4)$ at $F_v$
under $\xi$ has the $(3,1,1)$ cyclic-consecutive shape, with a block of
three consecutive positions $\{p, p+1, p+2\}$ (mod $5$) holding a common
colour $c$, and the remaining two positions $\{p+3, p+4\}$ holding the two
non-$c$ colours, one each. The algorithm's choice of $i_1$ forces
$\{i_1+3, i_1+4\}$ inside the $c$-block (so the chord is consistently
coloured) and the three positions $\{i_1, i_1+1, i_1+2\}$ pairwise distinct;
in particular $i_1+1$ lies \emph{outside} the $c$-block.
Choose $\varphi_1$ to be the proper $3$-edge-colouring of $H_1$ that agrees
with $\xi$ on every surviving edge and assigns each new edge at $A_j$ the
unique third colour at $A_j$. Then
$\varphi_1(\mathrm{spike}_1) = f_{i_1+1}$, a value not equal to $c$, while
$\varphi_1(\mathrm{merged}_1) = f_{i_1+3} = c$, so
$\varphi_1(\mathrm{spike}_1) \neq \varphi_1(\mathrm{merged}_1)$.
The same argument repeats at every step $k \geq 2$: the external vector at
the chosen pentagonal face under $\varphi_{k-1}$ has the $(3,1,1)$
cyclic-consecutive shape (Lemma~\ref{lem:pentagonal-externals}), the
algorithm's index choice $i_k$ puts $i_k+3, i_k+4$ inside the colour block
and $i_k+1$ outside, and step~(3d) thus assigns
$\varphi_k(\mathrm{spike}_k) \neq \varphi_k(\mathrm{merged}_k)$. The
algorithm preserves these colours through every later step, so
$\varphi_{t^*}(\mathrm{spike}_t) \neq \varphi_{t^*}(\mathrm{merged}_t)$ for
every $t \in \{1, \dots, t^*\}$.
\end{proof}
\begin{conjecture}
\label{conj:face-monochromatic-pair-on-merged-kempe-cycle}
Let $G$ be a minimal counterexample to the Four Colour Theorem, and let
$\widehat{G}'_{v,i}$ be a reduced dual of $G' = \mathrm{dual}(G)$. Then for
every proper $3$-edge-colouring $\varphi$ of $\widehat{G}'_{v,i}$ there exist
a face $F$ of $\widehat{G}'_{v,i}$ and two distinct edges
$e_1, e_2 \in \partial F$, with neither $e_1$ nor $e_2$ equal to the merged
edge, such that
\begin{enumerate}
\item $\varphi(e_1) = \varphi(e_2)$,
\item $e_1$, $e_2$, and the merged edge all lie on a common
$\{a, b\}$-Kempe cycle of $\varphi$, and
\item exactly one edge of $\partial F$ lies between $e_1$ and $e_2$ along
one of the two arcs of $\partial F$; equivalently, subdividing $e_1$
and $e_2$ by new vertices $X_1, X_2$ and joining them by a new edge
$X_1 X_2$ inside $F$ creates a new face bounded by exactly $4$
edges (the new edge, the two subdivision halves adjacent to it, and
the single $\partial F$-edge between $e_1$ and $e_2$).
\end{enumerate}
\end{conjecture}
\begin{remark}
\label{rem:conj-3-6-empirical}
The conjecture cannot be tested on actual minimal counterexamples (none exist
by the Four Colour Theorem), but its conclusion is checkable on the
structural surrogates: proper $3$-edge-colourings of reduced duals that
satisfy both the chord-apex condition (Lemma~\ref{lem:chord-apex}) and the
Kempe-cycle conditions (Lemma~\ref{lem:kempe-spike}), since a counterexample's
reduced dual is forced to admit such colourings under any proper colouring.
For every min-degree-$5$ triangulation $G$ with $|V(G)| \leq 21$, every
pentagonal face $F$ of $G'$, and every reduction index
$i \in \{0,\dots,4\}$, we enumerated all such colourings and tested the
three clauses of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
(see \texttt{experiments/check\_conj\_face\_kempe\_scaled.py}); $n = 22$
ran past a $1800$\,s budget after $641{,}700$ colourings (all pass), but
did not finish the full set of $651$ triangulations:
\begin{center}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{r|r|r|r|l}
$n$ & \#tri & \#col.\ tested & \#sat.\ & status \\
\hline
$12$ & $1$ & $0$ & --- & vacuous (icosahedron) \\
$13$ & $0$ & --- & --- & no min-deg-$5$ tri \\
$14$ & $1$ & $216$ & $216$ & all pass \\
$15$ & $1$ & $0$ & --- & vacuous \\
$16$ & $3$ & $864$ & $864$ & all pass \\
$17$ & $4$ & $4{,}650$ & $4{,}650$ & all pass \\
$18$ & $12$ & $8{,}070$ & $8{,}070$ & all pass \\
$19$ & $23$ & $21{,}138$ & $21{,}138$ & all pass \\
$20$ & $73$ & $107{,}874$ & $107{,}874$ & all pass \\
$21$ & $192$ & $392{,}370$ & $392{,}370$ & all pass \\
$22$ (part.) & $651$ & $641{,}700$ & $641{,}700$ & timeout \\
\hline
total ($n \le 21$) & $311$ & $535{,}182$ & $535{,}182$ & \\
\end{tabular}
\end{center}
\noindent The vacuous rows ($n = 12, 15$) are those where the relevant
reduced duals admit no proper $3$-edge-colouring satisfying chord-apex +
both Kempe-cycle conditions, so the conjecture has no content there. On
every $(G, F, i, \varphi)$ with content, all three clauses of the
conjecture hold simultaneously.
\end{remark}
The next definition records a cubic-preserving analogue of edge contraction
which turns out --- under planar duality --- to coincide with simple-graph
contraction on the dual side. It will be the central tool in
We introduce a cubic-preserving analogue of edge contraction which turns
out --- under planar duality --- to coincide with simple-graph contraction
on the dual side. It will be the central tool in
Section~\ref{sec:toward-4ct} below, where we formulate a sufficient
condition for the Four Colour Theorem.
@@ -788,39 +518,110 @@ by the contraction).}
\label{fig:thm-cubic-contraction-4face}
\end{figure}
\section{The Four Colour Theorem from a strengthened conjecture}
\section{The face-monochromatic-pair conjecture and the Four Colour Theorem}
\label{sec:toward-4ct}
The next conjecture strengthens
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} by adding
a clause that arranges the new $4$-edge face $f_n$ to satisfy the hypotheses
of Theorem~\ref{thm:cubic-contraction-4face}. The strengthening, if true,
would imply the Four Colour Theorem.
The following conjecture identifies a structural property of every proper
$3$-edge-colouring of a reduced dual of a minimal counterexample. If true,
it implies the Four Colour Theorem via
Theorem~\ref{thm:cubic-contraction-4face}.
\begin{conjecture}[Strengthening of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}]
The reduced-dual construction of Definition~\ref{def:reduced-dual} can also
be applied \emph{iteratively}, producing a sequence $H_1, H_2, \dots$ of
progressively smaller cubic plane graphs and tracking an accumulating set
of ``protected'' edges; that iterated reduction, with its termination
analysis and structural lemmas about spike/merged pairs at each stage, is
the subject of a companion paper and is not used below.
\begin{conjecture}[Face-monochromatic-pair conjecture]
\label{conj:face-monochromatic-pair-on-merged-kempe-cycle}
Let $G$ be a minimal counterexample to the Four Colour Theorem, and let
$\widehat{G}'_{v,i}$ be a reduced dual of $G' = \mathrm{dual}(G)$. Then
for every proper $3$-edge-colouring $\varphi$ of $\widehat{G}'_{v,i}$
there exist a face $F$ of $\widehat{G}'_{v,i}$ and two distinct edges
$e_1, e_2 \in \partial F$, with neither $e_1$ nor $e_2$ equal to the
merged edge, such that:
\begin{enumerate}
\item $\varphi(e_1) = \varphi(e_2)$. Write
$a := \varphi(e_1) = \varphi(e_2)$.
\item $e_1$, $e_2$, and the merged edge all lie on a common
$\{a, b\}$-Kempe cycle of $\varphi$, for some colour $b \neq a$.
\item Exactly one edge of $\partial F$ lies between $e_1$ and $e_2$
along one of the two arcs of $\partial F$. Equivalently,
subdividing $e_1, e_2$ by new vertices $X_1, X_2$ and joining
them by a new edge $X_1 X_2$ inside $F$ creates a new face $f_n$
bounded by exactly $4$ edges (the new edge $X_1 X_2$, the two
subdivision halves adjacent to it, and the single
$\partial F$-edge between $e_1$ and $e_2$).
\end{enumerate}
\end{conjecture}
\begin{remark}
\label{rem:conj-3-6-empirical}
\sloppy
The conjecture cannot be tested on actual minimal counterexamples (none
exist by the Four Colour Theorem), but its conclusion is checkable on the
structural surrogates: proper $3$-edge-colourings of reduced duals that
satisfy both the chord-apex condition (Lemma~\ref{lem:chord-apex}) and
the Kempe-cycle conditions (Lemma~\ref{lem:kempe-spike}), since a minimal
counterexample's reduced dual is forced to admit such colourings. For
every min-degree-$5$ triangulation $G$ with $|V(G)| \leq 21$, every
pentagonal face $F$ of $G'$, and every reduction index
$i \in \{0,\dots,4\}$, we enumerated all such colourings and verified the
three clauses of
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} (see
\texttt{experiments/check\_conj\_face\_kempe\_scaled.py}); $n = 22$ ran
past a $1800$\,s budget after $641{,}700$ colourings (all pass) without
finishing the full set of $651$ triangulations.
\begin{center}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{r|r|r|r|l}
$n$ & \#tri & \#col.\ tested & \#sat. & status \\
\hline
$12$ & $1$ & $0$ & --- & vacuous (icosahedron) \\
$13$ & $0$ & --- & --- & no min-deg-$5$ tri \\
$14$ & $1$ & $216$ & $216$ & all pass \\
$15$ & $1$ & $0$ & --- & vacuous \\
$16$ & $3$ & $864$ & $864$ & all pass \\
$17$ & $4$ & $4{,}650$ & $4{,}650$ & all pass \\
$18$ & $12$ & $8{,}070$ & $8{,}070$ & all pass \\
$19$ & $23$ & $21{,}138$ & $21{,}138$ & all pass \\
$20$ & $73$ & $107{,}874$ & $107{,}874$ & all pass \\
$21$ & $192$ & $392{,}370$ & $392{,}370$ & all pass \\
$22$ (part.) & $651$ & $641{,}700$ & $641{,}700$ & timeout \\
\hline
total ($n \le 21$) & $311$ & $535{,}182$ & $535{,}182$ & \\
\end{tabular}
\end{center}
\noindent The vacuous rows ($n = 12, 15$) are those where the relevant
reduced duals admit no chord-apex+Kempe colourings, so the conjecture
has no content there.
\end{remark}
The following strengthening adds a fourth clause that arranges the new
$4$-face $f_n$ to satisfy the hypotheses of
Theorem~\ref{thm:cubic-contraction-4face}.
\begin{conjecture}[Strengthening of
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}]
\label{conj:face-monochromatic-pair-strengthened}
Let $G$, $\widehat{G}'_{v,i}$, $\varphi$ be as in
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}. Then
there exist $F$, $e_1$, $e_2$ satisfying clauses (1)--(3) of that
conjecture, and the following additional clause holds.
Let $X_1, X_2$ be the new vertices subdividing $e_1, e_2$, joined by a new
edge $X_1 X_2$ inside $F$; write $\widehat{G}'^{+}$ for the resulting
modified graph (which has $|V(\widehat{G}'_{v,i})|+2$ vertices and
$|E(\widehat{G}'_{v,i})|+3$ edges, is again cubic and plane, and admits a
proper $3$-edge-colouring). Let $\varphi'$ be the proper
$3$-edge-colouring of $\widehat{G}'^{+}$ obtained from $\varphi$ by
swapping the two colours along the (subdivided) $\{a, b\}$-Kempe cycle of
clause~(2) and assigning the new edge $X_1 X_2$ the remaining (third)
colour. In particular $\varphi'$ agrees with $\varphi$ on every edge of
$\widehat{G}'_{v,i}$ outside that Kempe cycle, and at $X_1$ and $X_2$ the
two subdivision halves take the colours $\{a, b\}$ in the order forced by
propriety. Write $a := \varphi(e_1) = \varphi(e_2)$,
$c := \varphi'(X_1 X_2)$, and let $b$ be the third colour. Let $f_n$ be
the new $4$-edge face of $\widehat{G}'^{+}$ incident to $X_1 X_2$. Then:
there exist $F$, $e_1$, $e_2$ satisfying clauses~(1)--(3) of that
conjecture, together with the following additional clause.
\begin{enumerate}
\setcounter{enumi}{3}
\item either
\item Let $\widehat{G}'^{+}$ be the modified graph obtained from
$\widehat{G}'_{v,i}$ by the modification of clause~(3), and let
$\varphi'$ be the proper $3$-edge-colouring of $\widehat{G}'^{+}$
obtained from $\varphi$ by swapping the two colours along the
(subdivided) $\{a, b\}$-Kempe cycle of clause~(2) and assigning
the new edge $X_1 X_2$ the remaining (third) colour $c$.
(Equivalently: $\varphi'$ agrees with $\varphi$ on every edge of
$\widehat{G}'_{v,i}$ outside that Kempe cycle, and at $X_1, X_2$
the two subdivision halves take the colours $\{a, b\}$ in the
order forced by propriety.) Then either
\begin{enumerate}
\item[(i)] $\partial f_n$ uses all three colours under
$\varphi'$, or
@@ -836,30 +637,29 @@ the new $4$-edge face of $\widehat{G}'^{+}$ incident to $X_1 X_2$. Then:
\sloppy
The strengthened conjecture was tested on the same chord-apex+Kempe
colourings as Remark~\ref{rem:conj-3-6-empirical}; for each colouring we
sought any Conjecture-3.6-witness $(F, e_1, e_2)$ whose accompanying
$f_n$ satisfies clause~(4) (see
sought any
Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}-witness
$(F, e_1, e_2)$ whose accompanying $f_n$ satisfies clause~(4) (see
\texttt{experiments/check\_conj\_3\_8\_scaled.py}):
\begin{center}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{r|r|r|r|l}
$n$ & \#tri & \#col.\ tested & \#sat.\ & status \\
$n$ & \#tri & \#col.\ tested & \#sat. & status \\
\hline
$12$ & $1$ & $0$ & --- & vacuous \\
$13$ & $0$ & --- & --- & no min-deg-$5$ tri \\
$14$ & $1$ & $216$ & $216$ & all pass \\
$15$ & $1$ & $0$ & --- & vacuous \\
$16$ & $3$ & $864$ & $864$ & all pass \\
$17$ & $4$ & $4{,}650$ & $4{,}650$ & all pass \\
$18$ & $12$ & $8{,}070$ & $8{,}070$ & all pass \\
$14$ & $1$ & $216$ & $216$ & all pass \\
$16$ & $3$ & $864$ & $864$ & all pass \\
$17$ & $4$ & $4{,}650$ & $4{,}650$ & all pass \\
$18$ & $12$ & $8{,}070$ & $8{,}070$ & all pass \\
\hline
total & $23$ & $13{,}800$ & $13{,}800$ & \\
total ($n \le 18$) & $20$ & $13{,}800$ & $13{,}800$ & \\
\end{tabular}
\end{center}
\noindent A subtlety: only about half of the Conjecture-3.6-witnesses
individually satisfy clause (4) on each colouring, but in every case some
witness does. The conjecture is therefore an existential statement at the
witness level, not a property of every witness.
\noindent A subtlety: only about half of the
Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}-witnesses
individually satisfy clause~(4) on each colouring, but in every case
\emph{some} witness does. Clause~(4) is therefore an existential statement
at the witness level, not a property of every witness.
\end{remark}
\begin{remark}[The implication to the Four Colour Theorem]