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. LaTeX Warning: `h' float specifier changed to `ht'. [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). PDF statistics: - 192 PDF objects out of 1000 (max. 8388607) - 106 compressed objects within 2 object streams + 195 PDF objects out of 1000 (max. 8388607) + 108 compressed objects within 2 object streams 0 named destinations out of 1000 (max. 500000) 51 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 40e1d2e..5093106 100644 Binary files a/papers/face_monochromatic_pairs/paper.pdf and b/papers/face_monochromatic_pairs/paper.pdf differ diff --git a/papers/face_monochromatic_pairs/paper.tex b/papers/face_monochromatic_pairs/paper.tex index f0be990..276a30b 100644 --- a/papers/face_monochromatic_pairs/paper.tex +++ b/papers/face_monochromatic_pairs/paper.tex @@ -742,6 +742,59 @@ cannot be realised at $e$.} \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