diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.aux b/papers/dual_decomposition_minimal_counterexamples/paper.aux index 7fd54d6..2c8b4dc 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.aux +++ b/papers/dual_decomposition_minimal_counterexamples/paper.aux @@ -12,9 +12,10 @@ \newlabel{lem:chord-apex}{{2.6}{4}} \@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}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\gdef \@abspage@last{6} +\gdef \@abspage@last{7} diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.log b/papers/dual_decomposition_minimal_counterexamples/paper.log index ce6f332..68afba2 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) 22 MAY 2026 20:45 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 23 MAY 2026 02:40 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -226,50 +226,50 @@ uced_dual_step3.png> <./fig_reduced_dual_step4.png>] File: fig_chord_apex_step1.png Graphic file (type png) -Package pdftex.def Info: fig_chord_apex_step1.png used on input line 286. +Package pdftex.def Info: fig_chord_apex_step1.png used on input line 287. 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[4] [5 <./fig_chord_apex_step1.png> <./fig_chord_apex_step2.png> <./fig_chord_a -pex_step3.png>] [6] (./paper.aux) ) +pex_step3.png>] [6] [7] (./paper.aux) ) Here is how much of TeX's memory you used: - 3042 strings out of 478268 - 43139 string characters out of 5846347 - 343136 words of memory out of 5000000 - 21083 multiletter control sequences out of 15000+600000 + 3043 strings out of 478268 + 43156 string characters out of 5846347 + 343146 words of memory out of 5000000 + 21084 multiletter control sequences out of 15000+600000 476364 words of font info for 55 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 69i,8n,76p,664b,298s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (6 pages, 750077 bytes). + +Output written on paper.pdf (7 pages, 755306 bytes). PDF statistics: - 103 PDF objects out of 1000 (max. 8388607) - 54 compressed objects within 1 object stream + 108 PDF objects out of 1000 (max. 8388607) + 58 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 36 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 188ee4a..f1cef5f 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 627a8c3..fc39518 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.tex +++ b/papers/dual_decomposition_minimal_counterexamples/paper.tex @@ -177,11 +177,12 @@ $\widehat{G}'_{v,0}$.} \begin{definition}[Edges of the reduced dual] \label{def:edge-names} The four edges added in steps (3) and (4) of Definition~\ref{def:reduced-dual} -are named as follows. The chord $A_{i+3}A_{i+4}$ is the \emph{merged edge}; the -edge $A_{i+1}v_n$ is the \emph{spike edge}; and the edges $A_iv_n$ and -$A_{i+2}v_n$ are the \emph{side edges}. In the $i = 0$ case of -Figure~\ref{fig:reduced-dual-steps} these are $\{A_3, A_4\}$, $\{A_1, v_n\}$, -and $\{A_0, v_n\}, \{A_2, v_n\}$ respectively. +are named as follows. The chord $A_{i+3}A_{i+4}$ is the \emph{merged edge}; +the edge $A_{i+1}v_n$ is the \emph{spike edge}; the edge $A_iv_n$ is the +\emph{side-$0$ edge}; and the edge $A_{i+2}v_n$ is the \emph{side-$1$ edge}. +In the $i = 0$ case of Figure~\ref{fig:reduced-dual-steps} these are +$\{A_3, A_4\}$, $\{A_1, v_n\}$, $\{A_0, v_n\}$, and $\{A_2, v_n\}$ +respectively. \end{definition} We will use the following structural fact about proper $3$-edge-colourings near @@ -348,4 +349,46 @@ $G'$ is $3$-edge-colourable iff $G$ is $4$-vertex-colourable, contradicting that $G$ is a counterexample. The assumption $W \neq Y$ is therefore false. \end{proof} +For a pair of colours $\{a, b\} \subseteq \{1, 2, 3\}$, the subgraph of +$\widehat{G}'_{v,i}$ on the edges coloured $a$ or $b$ is $2$-regular (since at +each vertex exactly one of the three incident edges is excluded), and hence a +disjoint union of cycles. We call each such cycle a \emph{$\{a, b\}$-Kempe +cycle}, and reserve the notation for the specific cycle containing a given +edge when the context makes it clear. Swapping the two colours on a single +Kempe cycle yields another proper $3$-edge-colouring of the same graph. + +\begin{lemma}[Kempe cycles through the spike] +\label{lem:kempe-spike} +Let $G$ be a minimal counterexample, fix a reduced dual $\widehat{G}'_{v,i}$ of +$G'$, and let $\varphi$ be a proper $3$-edge-colouring of $\widehat{G}'_{v,i}$. +Write $c$ for the common colour assigned by $\varphi$ to the spike and the +merged edge (Lemma~\ref{lem:chord-apex}), and $c_0, c_1$ for the colours of +the side-$0$ and side-$1$ edges respectively, so $\{c, c_0, c_1\} = \{1, 2, +3\}$. Then +\begin{enumerate} + \item the $\{c, c_0\}$-Kempe cycle through the spike edge contains both the + side-$0$ edge and the merged edge; + \item the $\{c, c_1\}$-Kempe cycle through the spike edge contains both the + side-$1$ edge and the merged edge. +\end{enumerate} +\end{lemma} + +\begin{proof} +We prove (1); (2) is the same argument with $c_1$ and the side-$1$ edge in +place of $c_0$ and the side-$0$ edge. + +The spike edge $\{A_{i+1}, v_n\}$ and the side-$0$ edge $\{A_i, v_n\}$ share +the vertex $v_n$ and receive the two colours $c, c_0$, so they both lie on the +$\{c, c_0\}$-Kempe cycle through $v_n$. Suppose for contradiction that the +merged edge lies on a different $\{c, c_0\}$-Kempe cycle $K$ (it lies on +\emph{some} such cycle, since it has colour $c$). Let $\varphi'$ be obtained +from $\varphi$ by swapping the colours $c$ and $c_0$ along $K$ alone: this is +a Kempe swap, so $\varphi'$ is again a proper $3$-edge-colouring of +$\widehat{G}'_{v,i}$. Under $\varphi'$ the spike edge --- which is not on $K$ +--- still has colour $c$, but the merged edge --- which is on $K$ --- now has +colour $c_0$. Hence in $\varphi'$ the spike and the merged edge receive +distinct colours, contradicting Lemma~\ref{lem:chord-apex} applied to +$\varphi'$. +\end{proof} + \end{document}