\relax \citation{AH77a} \citation{AHK77} \citation{RSST97} \citation{Gonthier08} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \newlabel{sec:intro}{{1}{1}} \@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Organization.}}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Companion paper.}}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The minimal counterexample}}{2}{}\protected@file@percent } \newlabel{sec:minimal}{{2}{2}} \newlabel{lem:triangulate}{{2.1}{2}} \newlabel{def:minimal}{{2.2}{2}} \citation{Heawood1898} \newlabel{lem:mindeg}{{2.4}{3}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{The reduced dual}}{3}{}\protected@file@percent } \newlabel{sec:reduced-dual}{{3}{3}} \newlabel{def:heawood-number}{{3.1}{3}} \newlabel{def:reduced-dual}{{3.2}{4}} \newlabel{def:edge-names}{{3.4}{4}} \newlabel{lem:pentagonal-externals}{{3.5}{4}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 3.2\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{5}{}\protected@file@percent } \newlabel{fig:reduced-dual-steps}{{1}{5}} \newlabel{lem:chord-apex}{{3.7}{6}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 3.7\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 3.5\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{7}{}\protected@file@percent } \newlabel{fig:chord-apex-proof}{{2}{7}} \newlabel{lem:kempe-spike}{{3.8}{8}} \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Edge suppression}}{8}{}\protected@file@percent } \newlabel{sec:edge-suppression}{{4}{8}} \newlabel{def:edge-suppression}{{4.1}{8}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Edge suppression (Definition\nonbreakingspace 4.1\hbox {}). Left: a fragment of a cubic plane graph with the suppressed 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.}}{9}{}\protected@file@percent } \newlabel{fig:edge-suppression}{{3}{9}} \newlabel{thm:edge-suppression-4face}{{4.2}{9}} \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 4.2\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 suppressed 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 suppression).}}{10}{}\protected@file@percent } \newlabel{fig:thm-edge-suppression-4face}{{4}{10}} \@writefile{toc}{\contentsline {section}{\tocsection {}{5}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{10}{}\protected@file@percent } \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{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}} \citation{Heawood1898} \newlabel{cor:single-cycle-non-constancy}{{5.4}{13}} \newlabel{conj:no-two-constant-kempe-cycles}{{5.5}{13}} \newlabel{rem:no-two-constant-kempe-cycles-counterexample}{{5.6}{13}} \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Smallest counterexample to Conjecture\nonbreakingspace 5.5\hbox {}: a $C_{28}$ fullerene-style cubic plane graph (12 pentagons + 4 hexagons) with a proper $3$-edge-colouring on which $h_\varphi $ is simultaneously constant ($\equiv -1$) on the red/blue $12$-cycle and the red/green $12$-cycle, which share the colour-red edge $(0, 1)$. Light-shaded nodes are on $V(K_0) \cap V(K_1)$; medium-shaded on $V(K_0) \cup V(K_1) \setminus V(K_0) \cap V(K_1)$; grey on neither.}}{14}{}\protected@file@percent } \newlabel{fig:no-two-constant-kempe-counterexample}{{6}{14}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A reduction of Conjecture 5.1\hbox {} via Heawood's face-sum identity}}{14}{}\protected@file@percent } \newlabel{eq:heawood-face-sum}{{5.1}{14}} \newlabel{conj:deciding-face}{{5.7}{14}} \newlabel{thm:deciding-face-implies-conj-5-1}{{5.8}{14}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A partial structural proof of Conjecture\nonbreakingspace 5.7\hbox {}}}{15}{}\protected@file@percent } \newlabel{def:flank-face}{{5.9}{15}} \newlabel{lem:flank-length}{{5.10}{15}} \newlabel{lem:flank-covering-base}{{5.11}{15}} \newlabel{lem:flank-covering-hex}{{5.12}{16}} \newlabel{thm:deciding-face-partial}{{5.13}{17}} \newlabel{rem:n-i-6-flank-fails}{{5.14}{17}} \newlabel{def:outer-face}{{5.15}{17}} \newlabel{lem:outer-face-length}{{5.16}{17}} \newlabel{lem:outer-face-covering-base}{{5.17}{17}} \newlabel{thm:deciding-face-partial-extended}{{5.18}{18}} \newlabel{rem:deciding-face-remaining-case}{{5.19}{18}} \newlabel{conj:gprime-pentagon-fallback}{{5.20}{18}} \newlabel{lem:gprime-pigeonhole}{{5.21}{18}} \newlabel{rem:gprime-pigeonhole-stop}{{5.22}{18}} \citation{AH77a} \citation{AHK77} \citation{RSST97} \citation{Gonthier08} \newlabel{rem:deciding-face-empirical}{{5.23}{20}} \newlabel{rem:heawood-empirical}{{5.24}{20}} \newlabel{rem:conj-3-6-empirical}{{5.25}{21}} \newlabel{conj:face-monochromatic-pair-strengthened}{{5.26}{22}} \newlabel{rem:conj-3-8-empirical}{{5.27}{22}} \bibcite{Heawood1898}{1} \bibcite{AH77a}{2} \bibcite{AHK77}{3} \bibcite{RSST97}{4} \bibcite{Gonthier08}{5} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{14.69437pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} \newlabel{rem:implication-4ct}{{5.28}{23}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{23}{}\protected@file@percent } \gdef \@abspage@last{23}