19 lines
2.4 KiB
TeX
19 lines
2.4 KiB
TeX
\relax
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Kempe's Proof (Valid Portion)}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.1}{Setup}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.2}{Every Planar Graph Has a Vertex of Degree at Most 5}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.3}{Cases of Degree at Most 3}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.4}{Case of Degree 4}}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Resolution of Degree 5 Case}}{2}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The neighborhood of a degree-5 vertex $v_0$: a 5-cycle on $a_0,a_1,a_2,a_3,v_1$ with $v_0$ adjacent to each.}}{2}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The neighborhood of $v_0$ and $v_1$ in $G^*$: $G$ with the edge $\{v_0, v_1\}$ deleted, so the spoke to $v_1$ is absent.}}{3}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The neighborhood of $v_0$ and $v_1$ in $G'$: $G^*$ with $a_0$ and $a_3$ merged into $a_{03}$.}}{3}{}\protected@file@percent }
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent0}{0pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{29.38873pt}
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\newlabel{tocindent3}{0pt}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces A proper 4-coloring of $G'$ with $\phi (v_0)=\phi (v_1)=\text {red}$, $\phi (a_{03})=\text {blue}$, $\phi (a_1)=\text {green}$, $\phi (a_2)=\text {yellow}$. By the previous lemma, $v_0$ and $v_1$ must lie in both the $\{\text {red},\text {green}\}$-Kempe chain (dashed green, passing through $a_1$) and the $\{\text {red},\text {yellow}\}$-Kempe chain (dashed yellow, passing through $a_2$).}}{4}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The same coloring lifted to $G^*$: $\phi (v_0)=\phi (v_1)=\text {red}$, $\phi (a_0)=\phi (a_3)=\text {blue}$, $\phi (a_1)=\text {green}$, $\phi (a_2)=\text {yellow}$. The $\{\text {red},\text {green}\}$- and $\{\text {red},\text {yellow}\}$-Kempe chains connecting $v_0$ to $v_1$ pass through $a_1$ and $a_2$ respectively.}}{4}{}\protected@file@percent }
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\gdef \@abspage@last{5}
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