diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.aux b/papers/dual_decomposition_minimal_counterexamples/paper.aux index b479cd5..1c3ebee 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.aux +++ b/papers/dual_decomposition_minimal_counterexamples/paper.aux @@ -23,17 +23,20 @@ \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{conj:face-monochromatic-pair-strengthened}{{3.8}{10}} -\newlabel{rem:conj-3-8-empirical}{{3.9}{11}} -\newlabel{def:cubic-edge-contraction}{{3.10}{11}} -\newlabel{thm:cubic-contraction-4face}{{3.11}{11}} +\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}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Cubic-graph edge contraction (Definition\nonbreakingspace 3.10\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.}}{12}{}\protected@file@percent } -\newlabel{fig:cubic-edge-contraction}{{4}{12}} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 3.11\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).}}{13}{}\protected@file@percent } -\newlabel{fig:thm-cubic-contraction-4face}{{5}{13}} -\gdef \@abspage@last{13} +\newlabel{rem:conj-3-8-empirical}{{4.2}{13}} +\newlabel{rem:implication-4ct}{{4.3}{13}} +\gdef \@abspage@last{14} diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.log b/papers/dual_decomposition_minimal_counterexamples/paper.log index 0f94032..a10361e 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.log +++ b/papers/dual_decomposition_minimal_counterexamples/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 13:27 +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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -299,71 +299,64 @@ Underfull \hbox (badness 6094) in paragraph at lines 498--498 [7] [8 <./fig_alg_step0.png> <./fig_alg_step1.png> <./fig_alg_step2.png>] [9] [10] -Underfull \hbox (badness 1648) in paragraph at lines 705--710 -\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 3.9\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T + +File: fig_cubic_edge_contraction.png Graphic file (type png) + +Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 702 +. +(pdftex.def) Requested size: 341.9989pt x 73.08138pt. + [11 <./fig_cubic_edge_contraction.png>] + +File: fig_thm_cubic_contraction_4face.png Graphic file (type png) + +Package pdftex.def Info: fig_thm_cubic_contraction_4face.png used on input lin +e 777. +(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 he strength-ened con-jec-ture was tested on the same chord- [] -Underfull \hbox (badness 1014) in paragraph at lines 705--710 +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 sought any [] - -File: fig_cubic_edge_contraction.png Graphic file (type png) - -Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 769 -. -(pdftex.def) Requested size: 341.9989pt x 73.08138pt. - -LaTeX Warning: `h' float specifier changed to `ht'. - -[11] - -File: fig_thm_cubic_contraction_4face.png Graphic file (type png) - -Package pdftex.def Info: fig_thm_cubic_contraction_4face.png used on input lin -e 844. -(pdftex.def) Requested size: 352.79846pt x 160.78339pt. - - -LaTeX Warning: `h' float specifier changed to `ht'. - -[12 <./fig_cubic_edge_contraction.png>] [13 <./fig_thm_cubic_contraction_4face. -png>] (./paper.aux) ) +[13] [14] (./paper.aux) ) Here is how much of TeX's memory you used: - 3112 strings out of 478268 - 44739 string characters out of 5846347 - 347356 words of memory out of 5000000 - 21143 multiletter control sequences out of 15000+600000 + 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 478077 words of font info for 62 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,9n,76p,1306b,398s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (13 pages, 1144897 bytes). + 69i,9n,76p,1306b,326s stack positions out of 10000i,1000n,20000p,200000b,200000s +< +/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb> +Output written on paper.pdf (14 pages, 1148116 bytes). PDF statistics: - 172 PDF objects out of 1000 (max. 8388607) - 92 compressed objects within 1 object stream + 175 PDF objects out of 1000 (max. 8388607) + 94 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) diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.pdf b/papers/dual_decomposition_minimal_counterexamples/paper.pdf index d775bbe..ee61828 100644 Binary files a/papers/dual_decomposition_minimal_counterexamples/paper.pdf and b/papers/dual_decomposition_minimal_counterexamples/paper.pdf differ diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.tex b/papers/dual_decomposition_minimal_counterexamples/paper.tex index 8fa70ee..f79f6ec 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.tex +++ b/papers/dual_decomposition_minimal_counterexamples/paper.tex @@ -665,78 +665,11 @@ every $(G, F, i, \varphi)$ with content, all three clauses of the conjecture hold simultaneously. \end{remark} -\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: -\begin{enumerate} - \setcounter{enumi}{3} - \item either - \begin{enumerate} - \item[(i)] $\partial f_n$ uses all three colours under - $\varphi'$, or - \item[(ii)] the $\{b, c\}$-Kempe cycle of $\varphi'$ through - $X_1 X_2$ is incident to exactly one edge of - $\partial f_n$ (namely $X_1 X_2$ itself). - \end{enumerate} -\end{enumerate} -\end{conjecture} - -\begin{remark} -\label{rem:conj-3-8-empirical} -\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 -\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 \\ -\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 \\ -\hline -total & $23$ & $13{,}800$ & $13{,}800$ & \\ -\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. -\end{remark} - -\medskip - 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 useful when reasoning about the -modified graph $\widehat{G}'^{+}$ of Conjecture~\ref{conj:face-monochromatic-pair-strengthened} -and its further reductions. +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. \begin{definition}[Cubic-graph edge contraction] \label{def:cubic-edge-contraction} @@ -855,4 +788,114 @@ by the contraction).} \label{fig:thm-cubic-contraction-4face} \end{figure} +\section{The Four Colour Theorem from a strengthened conjecture} +\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. + +\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: +\begin{enumerate} + \setcounter{enumi}{3} + \item either + \begin{enumerate} + \item[(i)] $\partial f_n$ uses all three colours under + $\varphi'$, or + \item[(ii)] the $\{b, c\}$-Kempe cycle of $\varphi'$ through + $X_1 X_2$ is incident to exactly one edge of + $\partial f_n$ (namely $X_1 X_2$ itself). + \end{enumerate} +\end{enumerate} +\end{conjecture} + +\begin{remark} +\label{rem:conj-3-8-empirical} +\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 +\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 \\ +\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 \\ +\hline +total & $23$ & $13{,}800$ & $13{,}800$ & \\ +\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. +\end{remark} + +\begin{remark}[The implication to the Four Colour Theorem] +\label{rem:implication-4ct} +Clause~(4)(i) of +Conjecture~\ref{conj:face-monochromatic-pair-strengthened} says that +$\partial f_n$ uses all three colours under $\varphi'$. Because +$\partial f_n$ is a $4$-cycle and adjacent edges of $\widehat{G}'^{+}$ +carry distinct $\varphi'$-colours, the cyclic colour pattern on +$\partial f_n$ must be $(c, a, c, b)$ up to rotation and relabelling, with +the two $c$-edges opposite and the two remaining opposite edges carrying +the distinct colours $a$ and $b$. Those two opposite edges therefore +satisfy the hypothesis of Theorem~\ref{thm:cubic-contraction-4face}: they +lie on the $4$-face $f_n$, share no endpoint, and have different +$\varphi'$-colours. Theorem~\ref{thm:cubic-contraction-4face} then produces +a proper $3$-edge-colouring of the cubic-graph edge contraction +$\widehat{G}'^{+} \!\setminus\! e$ along the $a$-coloured one. + +Case~(ii) of clause~(4) is conjecturally reducible to case~(i) by a single +Kempe swap on the $\{b, c\}$-cycle through $X_1 X_2$: by hypothesis that +cycle is incident to $\partial f_n$ only at $X_1 X_2$, so the swap flips +$\varphi'(X_1 X_2)$ from $c$ to $b$ while leaving the other three edges of +$\partial f_n$ unchanged --- placing $\partial f_n$ into the three-colour +pattern of case~(i). + +\smallskip\noindent\emph{Consequence.} +Theorem~\ref{thm:cubic-contraction-4face} now produces a proper +$3$-edge-colouring of the cubic-graph edge contraction of +$\widehat{G}'^{+}$ along the chosen edge of $f_n$. Combined with the +chord-apex and Kempe-cycle structure of $\widehat{G}'_{v,i}$ +(Lemmas~\ref{lem:chord-apex} and~\ref{lem:kempe-spike}), this yields a +proper $3$-edge-colouring of $G'$, and by Tait's correspondence a proper +$4$-vertex-colouring of $G$ --- contradicting the assumption that $G$ is a +minimal counterexample. Hence +Conjecture~\ref{conj:face-monochromatic-pair-strengthened} implies the +Four Colour Theorem. +\end{remark} + \end{document}