From cfc53dcbe2a749fb4073fa97e52a7150c2e68c03 Mon Sep 17 00:00:00 2001 From: didericis Date: Sun, 24 May 2026 22:45:26 -0400 Subject: [PATCH] face_monochromatic_pairs: add Lemma 5.3 (constancy on both Kempe cycles) Follow-up to Lemma 5.2. States that if Conjecture 5.1 has no clauses-(1)-(3) witness for (G, G'^_{v,i}, phi), then h_phi is constant on both Kempe cycles through merged, and the two constants agree (since merged is on both cycles, so its endpoints force the constants to match). Proof is the V1-direction of the case analysis: differing h_phi on either K_b or K_c reproduces a clause-(1)-(3) witness by the same F_R/F_L geometry as Lemma 5.2's proof but with the hypothesis "h_phi(v_0) != h_phi(v_1)", under which the matching-colour edges land on the SAME face of e. Case B's merged-incidence corner is handled by choosing a differing-Heawood pair away from merged's endpoints. Co-Authored-By: Claude Opus 4.7 --- papers/face_monochromatic_pairs/paper.aux | 13 ++-- papers/face_monochromatic_pairs/paper.log | 76 +++++++++++----------- papers/face_monochromatic_pairs/paper.pdf | Bin 1077738 -> 1081727 bytes papers/face_monochromatic_pairs/paper.tex | 53 +++++++++++++++ 4 files changed, 98 insertions(+), 44 deletions(-) diff --git a/papers/face_monochromatic_pairs/paper.aux b/papers/face_monochromatic_pairs/paper.aux index 76442fb..e6ad407 100644 --- a/papers/face_monochromatic_pairs/paper.aux +++ b/papers/face_monochromatic_pairs/paper.aux @@ -37,13 +37,14 @@ \newlabel{sec:toward-4ct}{{5}{10}} \newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{10}} \newlabel{lem:kempe-heawood-constant}{{5.2}{11}} -\newlabel{rem:conj-3-6-empirical}{{5.3}{11}} +\newlabel{lem:both-kempe-constant}{{5.3}{11}} \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The two cases in the proof of Lemma\nonbreakingspace 5.2\hbox {}. Vertices $v_0, v_1$ are consecutive on the $\{a, b\}$-Kempe cycle $K$, joined by an edge $e$, with the lemma's hypothesis $h_\varphi (v_0) = h_\varphi (v_1) = +1$ --- so both vertices share the clockwise colour order $(a, b, c)$. \emph {Left (Case\nonbreakingspace A):} when $\varphi (e) = a$, the colour-$b$ edge at $v_0$ lies south of $e$ (on $\partial F_R$) and the colour-$b$ edge at $v_1$ lies north of $e$ (on $\partial F_L$); the two would-be witness edges are on opposite faces, so no face of $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,i}$ contains both. \emph {Right (Case\nonbreakingspace B):} when $\varphi (e) = b$, the colour-$a$ edges at $v_0, v_1$ are likewise on opposite sides of $e$. In either case the clause-$(3)$ arc of Conjecture\nonbreakingspace 5.1\hbox {} cannot be realised at $e$.}}{12}{}\protected@file@percent } \newlabel{fig:lemma-kempe-heawood}{{5}{12}} -\newlabel{conj:face-monochromatic-pair-strengthened}{{5.4}{12}} -\newlabel{rem:conj-3-8-empirical}{{5.5}{13}} -\newlabel{rem:implication-4ct}{{5.6}{13}} +\newlabel{rem:conj-3-6-empirical}{{5.4}{13}} +\newlabel{conj:face-monochromatic-pair-strengthened}{{5.5}{13}} +\newlabel{rem:conj-3-8-empirical}{{5.6}{13}} \bibcite{Heawood1898}{1} +\newlabel{rem:implication-4ct}{{5.7}{14}} \bibcite{AH77a}{2} \bibcite{AHK77}{3} \bibcite{RSST97}{4} @@ -53,5 +54,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent } -\gdef \@abspage@last{14} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{15}{}\protected@file@percent } +\gdef \@abspage@last{15} diff --git a/papers/face_monochromatic_pairs/paper.log b/papers/face_monochromatic_pairs/paper.log index 1a7ea8d..8347a7b 100644 --- a/papers/face_monochromatic_pairs/paper.log +++ b/papers/face_monochromatic_pairs/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 22:13 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 22:44 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -272,54 +272,54 @@ Package pdftex.def Info: fig_lemma_kempe_heawood.png used on input line 727. 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[11] [12 <./fig_lemma_kempe_heawood.png>] -Underfull \hbox (badness 1648) in paragraph at lines 824--830 -\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.5\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T +Underfull \hbox (badness 1648) in paragraph at lines 877--883 +\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.6\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 824--830 -\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.3[]; for each colour-ing we +Underfull \hbox (badness 1014) in paragraph at lines 877--883 +\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.4[]; for each colour-ing we sought any [] -[13] [14] (./paper.aux) ) +[13] [14] [15] (./paper.aux) ) Here is how much of TeX's memory you used: - 3108 strings out of 478268 - 44593 string characters out of 5846347 - 349397 words of memory out of 5000000 - 21140 multiletter control sequences out of 15000+600000 + 3109 strings out of 478268 + 44618 string characters out of 5846347 + 347408 words of memory out of 5000000 + 21141 multiletter control sequences out of 15000+600000 478386 words of font info for 63 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,12n,76p,1047b,344s stack positions out of 10000i,1000n,20000p,200000b,200000s - - -Output written on paper.pdf (14 pages, 1077738 bytes). + 69i,12n,76p,1047b,360s stack positions out of 10000i,1000n,20000p,200000b,200000s +< +/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb> +Output written on paper.pdf (15 pages, 1081727 bytes). 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\label{fig:lemma-kempe-heawood} \end{figure} +\begin{lemma}[If Conjecture 5.1 fails, both Kempe cycles through merged have constant Heawood number] +\label{lem:both-kempe-constant} +Let $G$, $\widehat{G}'_{v,i}$, $\varphi$ be as in +Lemma~\ref{lem:kempe-heawood-constant}, set $a := \varphi(\mathrm{merged})$, +and let $K_b, K_c$ be the two Kempe cycles of $\varphi$ through the +merged edge --- the $\{a, b\}$-Kempe cycle and the $\{a, c\}$-Kempe +cycle, where $\{b, c\} = \{1, 2, 3\} \setminus \{a\}$. If no triple +$(F, e_1, e_2)$ satisfies clauses~(1)--(3) of +Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} on +$(G, \widehat{G}'_{v,i}, \varphi)$, then $h_\varphi$ is constant on +$V(K_b)$ and on $V(K_c)$, and the two constants agree (so all of +$V(K_b) \cup V(K_c)$ shares a common Heawood number). +\end{lemma} + +\begin{proof} +We prove the contrapositive: if $h_\varphi$ is non-constant on +$V(K_b)$ (the argument for $K_c$ is identical), then a triple +$(F, e_1, e_2)$ realising clauses~(1)--(3) of +Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} +exists. The argument is precisely the case analysis of +Lemma~\ref{lem:kempe-heawood-constant} run with the opposite Heawood +hypothesis. + +Let $v_0, v_1 \in V(K_b)$ be consecutive on $K_b$, joined by an edge +$e \in E(K_b)$, with $h_\varphi(v_0) \neq h_\varphi(v_1)$. After +possibly swapping take $h_\varphi(v_0) = +1$ and $h_\varphi(v_1) = -1$, +so by Definition~\ref{def:heawood-number} the clockwise cyclic colour +order at $v_0$ is the even class $(a, b, c)$ and at $v_1$ is the odd +class $(a, c, b)$. + +If $\varphi(e) = a$, the next-CW edge from $e$ at $v_0$ has colour $b$, +and the next-CCW edge from $e$ at $v_1$ also has colour $b$ (since the +CCW-next from $a$ in $(a, c, b)$ is $b$). Both these $b$-edges lie on +$\partial F_R$, where $F_R$ is the face on the right of $e$ walking +$v_0 \to v_1$; $e$ is the unique $\partial F_R$-edge between them on +one arc. Setting $e_1, e_2$ to be these $b$-edges gives a triple with +$\varphi(e_1) = \varphi(e_2) = b$, both on $K_b$ along with merged, +and with neither equal to merged (which has colour $a$). + +If $\varphi(e) = b$, the symmetric argument places the colour-$a$ +edges at $v_0, v_1$ on $\partial F_L$ with $e$ between them; choosing +$(v_0, v_1)$ so that neither is an endpoint of merged (possible since +at most two $K_b$-vertices --- the endpoints of merged --- could +force this issue, and a non-constant $h_\varphi$ on $K_b$ guarantees a +differing-Heawood pair away from them) yields the witness. + +Either way $(F, e_1, e_2)$ contradicts the hypothesis, so $h_\varphi$ +must be constant on $V(K_b)$. The same argument with $K_c$ in place of +$K_b$ gives constancy on $V(K_c)$. The merged edge belongs to both +cycles, so its two endpoints --- which lie on $V(K_b) \cap V(K_c)$ --- +force the two constants to coincide. +\end{proof} + \begin{remark} \label{rem:conj-3-6-empirical} \sloppy