\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}} \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:reduced-dual}{{3.1}{3}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 3.1\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}$.}}{4}{}\protected@file@percent } \newlabel{fig:reduced-dual-steps}{{1}{4}} \newlabel{def:edge-names}{{3.3}{4}} \newlabel{lem:pentagonal-externals}{{3.4}{4}} \newlabel{lem:chord-apex}{{3.6}{5}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 3.6\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 3.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{6}{}\protected@file@percent } \newlabel{fig:chord-apex-proof}{{2}{6}} \newlabel{lem:kempe-spike}{{3.7}{7}} \@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.}}{8}{}\protected@file@percent } \newlabel{fig:edge-suppression}{{3}{8}} \newlabel{thm:edge-suppression-4face}{{4.2}{8}} \@writefile{toc}{\contentsline {section}{\tocsection {}{5}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{9}{}\protected@file@percent } \newlabel{sec:toward-4ct}{{5}{9}} \newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{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}} \newlabel{rem:conj-3-6-empirical}{{5.2}{10}} \newlabel{conj:face-monochromatic-pair-strengthened}{{5.3}{11}} \newlabel{rem:conj-3-8-empirical}{{5.4}{11}} \bibcite{AH77a}{1} \bibcite{AHK77}{2} \bibcite{RSST97}{3} \bibcite{Gonthier08}{4} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} \newlabel{rem:implication-4ct}{{5.5}{12}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent } \gdef \@abspage@last{12}