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 <noreply@anthropic.com>
This commit is contained in:
@@ -37,13 +37,14 @@
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\newlabel{sec:toward-4ct}{{5}{10}}
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\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{10}}
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\newlabel{lem:kempe-heawood-constant}{{5.2}{11}}
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\newlabel{rem:conj-3-6-empirical}{{5.3}{11}}
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\newlabel{lem:both-kempe-constant}{{5.3}{11}}
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\@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 }
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\newlabel{fig:lemma-kempe-heawood}{{5}{12}}
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\newlabel{conj:face-monochromatic-pair-strengthened}{{5.4}{12}}
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\newlabel{rem:conj-3-8-empirical}{{5.5}{13}}
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\newlabel{rem:implication-4ct}{{5.6}{13}}
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\newlabel{rem:conj-3-6-empirical}{{5.4}{13}}
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\newlabel{conj:face-monochromatic-pair-strengthened}{{5.5}{13}}
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\newlabel{rem:conj-3-8-empirical}{{5.6}{13}}
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\bibcite{Heawood1898}{1}
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\newlabel{rem:implication-4ct}{{5.7}{14}}
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\bibcite{AH77a}{2}
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\bibcite{AHK77}{3}
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\bibcite{RSST97}{4}
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he strength-ened con-jec-ture was tested on the same chord-
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[]
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@@ -742,6 +742,59 @@ cannot be realised at $e$.}
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\label{fig:lemma-kempe-heawood}
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\end{figure}
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\begin{lemma}[If Conjecture 5.1 fails, both Kempe cycles through merged have constant Heawood number]
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\label{lem:both-kempe-constant}
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Let $G$, $\widehat{G}'_{v,i}$, $\varphi$ be as in
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Lemma~\ref{lem:kempe-heawood-constant}, set $a := \varphi(\mathrm{merged})$,
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and let $K_b, K_c$ be the two Kempe cycles of $\varphi$ through the
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merged edge --- the $\{a, b\}$-Kempe cycle and the $\{a, c\}$-Kempe
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cycle, where $\{b, c\} = \{1, 2, 3\} \setminus \{a\}$. If no triple
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$(F, e_1, e_2)$ satisfies clauses~(1)--(3) of
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Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} on
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$(G, \widehat{G}'_{v,i}, \varphi)$, then $h_\varphi$ is constant on
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$V(K_b)$ and on $V(K_c)$, and the two constants agree (so all of
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$V(K_b) \cup V(K_c)$ shares a common Heawood number).
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\end{lemma}
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\begin{proof}
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We prove the contrapositive: if $h_\varphi$ is non-constant on
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$V(K_b)$ (the argument for $K_c$ is identical), then a triple
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$(F, e_1, e_2)$ realising clauses~(1)--(3) of
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Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
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exists. The argument is precisely the case analysis of
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Lemma~\ref{lem:kempe-heawood-constant} run with the opposite Heawood
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hypothesis.
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Let $v_0, v_1 \in V(K_b)$ be consecutive on $K_b$, joined by an edge
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$e \in E(K_b)$, with $h_\varphi(v_0) \neq h_\varphi(v_1)$. After
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possibly swapping take $h_\varphi(v_0) = +1$ and $h_\varphi(v_1) = -1$,
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so by Definition~\ref{def:heawood-number} the clockwise cyclic colour
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order at $v_0$ is the even class $(a, b, c)$ and at $v_1$ is the odd
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class $(a, c, b)$.
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If $\varphi(e) = a$, the next-CW edge from $e$ at $v_0$ has colour $b$,
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and the next-CCW edge from $e$ at $v_1$ also has colour $b$ (since the
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CCW-next from $a$ in $(a, c, b)$ is $b$). Both these $b$-edges lie on
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$\partial F_R$, where $F_R$ is the face on the right of $e$ walking
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$v_0 \to v_1$; $e$ is the unique $\partial F_R$-edge between them on
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one arc. Setting $e_1, e_2$ to be these $b$-edges gives a triple with
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$\varphi(e_1) = \varphi(e_2) = b$, both on $K_b$ along with merged,
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and with neither equal to merged (which has colour $a$).
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If $\varphi(e) = b$, the symmetric argument places the colour-$a$
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edges at $v_0, v_1$ on $\partial F_L$ with $e$ between them; choosing
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$(v_0, v_1)$ so that neither is an endpoint of merged (possible since
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at most two $K_b$-vertices --- the endpoints of merged --- could
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force this issue, and a non-constant $h_\varphi$ on $K_b$ guarantees a
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differing-Heawood pair away from them) yields the witness.
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Either way $(F, e_1, e_2)$ contradicts the hypothesis, so $h_\varphi$
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must be constant on $V(K_b)$. The same argument with $K_c$ in place of
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$K_b$ gives constancy on $V(K_c)$. The merged edge belongs to both
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cycles, so its two endpoints --- which lie on $V(K_b) \cap V(K_c)$ ---
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force the two constants to coincide.
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\end{proof}
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\begin{remark}
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\label{rem:conj-3-6-empirical}
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\sloppy
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