From ada26b680118a17cc2f1bf921185bd6052bce132 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 13 Apr 2026 11:01:31 -0400 Subject: [PATCH] chore: SAVEPOINT --- paper.aux | 19 ++++- paper.fdb_latexmk | 20 ++++-- paper.fls | 20 +++++- paper.log | 34 +++++---- paper.pdf | Bin 64345 -> 213600 bytes paper.synctex.gz | Bin 1625 -> 43824 bytes paper.tex | 176 ++++++++++++++++++++++++++++++++++++++++++++++ 7 files changed, 249 insertions(+), 20 deletions(-) diff --git a/paper.aux b/paper.aux index 05a181d..36a755f 100644 --- a/paper.aux +++ b/paper.aux @@ -1,7 +1,20 @@ \relax +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Kempe's Proof (Valid Portion)}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.1}{Setup}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.2}{Every Planar Graph Has a Vertex of Degree at Most 5}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.3}{Cases of Degree at Most 3}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{1.4}{Case of Degree 4}}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Resolution of Degree 5 Case}}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.1}{At Least 12 Vertices of Degree 5}}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.2}{Two Non-Adjacent Vertices of Degree 5}}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.3}{The Reduced Subgraph $G'$}}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.4}{Not All Colorings in $\Phi $ Saturate Both Neighborhoods}}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Merged Subgraphs of $G'$}}{3}{}\protected@file@percent } \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} -\newlabel{tocindent1}{0pt} -\newlabel{tocindent2}{0pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{29.38873pt} \newlabel{tocindent3}{0pt} -\gdef \@abspage@last{1} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Colorings of Merged Subgraphs Extend to $G'$}}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{4-Saturating Merged Colorings Extend to 4-Saturating Colorings of $G'$}}{4}{}\protected@file@percent } +\gdef \@abspage@last{4} diff --git a/paper.fdb_latexmk b/paper.fdb_latexmk index 3e26c02..61d2d8c 100644 --- a/paper.fdb_latexmk +++ b/paper.fdb_latexmk @@ -1,6 +1,6 @@ # Fdb version 3 -["pdflatex"] 1776046763 "/Users/didericis/Code/magnum-opus-4ct/paper.tex" "paper.pdf" "paper" 1776046763 - "/Users/didericis/Code/magnum-opus-4ct/paper.tex" 1776046763 2179 8055ca3cd1baeac7b377564e439027b4 "" +["pdflatex"] 1776052801 "/Users/didericis/Code/magnum-opus-4ct/paper.tex" "paper.pdf" "paper" 1776052801 + "/Users/didericis/Code/magnum-opus-4ct/paper.tex" 1776052800 13483 66b66eccf22ca3f14d4a88ccbcab3f5d "" "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -18,14 +18,26 @@ "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm" 1136768653 768 d7b9a2629a0c353102ad947dc9221d49 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb" 1248133631 31809 8670ca339bf94e56da1fc21c80635e2a "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb" 1248133631 32915 7bf7720c61a5b3a7ff25b0964421c9b6 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb" 1248133631 35660 fb24af7afbadb71801619f1415838111 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb" 1248133631 24287 6b803fa9eb1ddff9112e00519b09dd9e "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb" 1248133631 34694 ad62b13721ee8eda1dcc8993c8bd7041 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls" 1591045760 61881 a7369c346c2922a758ae6283cc1ed014 "" "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c "" "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd" 1359763108 961 6518c6525a34feb5e8250ffa91731cff "" @@ -40,8 +52,8 @@ "/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 "" "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" - "paper.aux" 1776046763 168 23aeeb56264b611b42959e54d96db7ab "pdflatex" - "paper.tex" 1776046763 2179 8055ca3cd1baeac7b377564e439027b4 "" + "paper.aux" 1776052801 1899 6f6c9f990e281a7651f9bbdc326c1739 "pdflatex" + "paper.tex" 1776052800 13483 66b66eccf22ca3f14d4a88ccbcab3f5d "" (generated) "paper.aux" "paper.log" diff --git a/paper.fls b/paper.fls index 90f7e41..d487f18 100644 --- a/paper.fls +++ b/paper.fls @@ -135,13 +135,31 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm -INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm OUTPUT paper.pdf INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm INPUT paper.aux INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt8.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb diff --git a/paper.log b/paper.log index e99ab96..a8233a2 100644 --- a/paper.log +++ b/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) 12 APR 2026 22:19 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 13 APR 2026 00:00 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -130,20 +130,30 @@ File: umsa.fd 2013/01/14 v3.01 AMS symbols A LaTeX Font Info: Trying to load font information for U+msb on input line 52. 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follows a \emph{minimal counterexample} argument. Suppose, for contradiction, that there exists a planar graph requiring 5 colors. Among all such graphs, let $G$ be one with the fewest vertices. Then every planar graph with fewer vertices than $G$ is 4-colorable, but $G$ itself is not. + +\subsection{Every Planar Graph Has a Vertex of Degree at Most 5} + +\begin{lemma} +Every planar graph has at least one vertex of degree $\leq 5$. +\end{lemma} + +\begin{proof} +Let $G$ be a connected planar graph with $V$ vertices, $E$ edges, and $F$ faces. By Euler's formula, +\[ + V - E + F = 2. +\] +Since every face is bounded by at least 3 edges and each edge borders at most 2 faces, we have $2E \geq 3F$, hence $F \leq \frac{2E}{3}$. Substituting into Euler's formula: +\[ + V - E + \frac{2E}{3} \geq 2 \implies E \leq 3V - 6. +\] +If every vertex had degree $\geq 6$, then $2E \geq 6V$, so $E \geq 3V$, contradicting $E \leq 3V - 6$. Therefore at least one vertex has degree $\leq 5$. +\end{proof} + +Since $G$ is a minimal counterexample, it must contain a vertex $v$ of degree at most 5. Kempe argued by cases on the degree of $v$. + +\subsection{Cases of Degree at Most 3} + +Suppose $v$ has degree $\leq 3$. Remove $v$ from $G$ to obtain the graph $G - v$. Since $G - v$ has fewer vertices than $G$, it is 4-colorable by minimality. Fix such a 4-coloring. Now reinsert $v$: its at most 3 neighbors occupy at most 3 of the 4 colors, so at least one color remains available for $v$. This yields a valid 4-coloring of $G$, a contradiction. + +\subsection{Case of Degree 4} + +Suppose $v$ has degree exactly 4 with neighbors $a$, $b$, $c$, $d$ appearing in cyclic order around $v$ in the planar embedding. Remove $v$ and 4-color $G - v$ by minimality. If the four neighbors do not all receive distinct colors, then at least one color is unused among them, and we may assign that color to $v$, giving a contradiction. + +So assume $a$, $b$, $c$, $d$ receive all four distinct colors; call them $1$, $2$, $3$, $4$ respectively. Define a \emph{Kempe chain} to be a maximal connected subgraph whose vertices are colored with exactly two specified colors. + +Consider the Kempe chain $K_{13}$ containing $a$ (using colors 1 and 3). + +\begin{itemize} + \item \textbf{Case 1}: $c$ is not in $K_{13}$. Swap colors 1 and 3 throughout $K_{13}$. This is still a valid coloring of $G - v$, and now $a$ receives color 3, so color 1 is free for $v$. + + \item \textbf{Case 2}: $c$ is in $K_{13}$. Then there is a path of alternating colors 1 and 3 from $a$ to $c$ in the planar embedding. Because $a$ and $c$ alternate around $v$ with $b$ and $d$, this path separates $b$ from $d$ in the plane. Therefore $b$ and $d$ lie in different Kempe chains for colors 2 and 4. Swap colors 2 and 4 in the chain containing $b$; now $b$ receives color 4, freeing color 2 for $v$. +\end{itemize} + +In both cases we obtain a valid 4-coloring of $G$, a contradiction. + +\section{Resolution of Degree 5 Case} + +\subsection{At Least 12 Vertices of Degree 5} + +Since we have already handled every vertex of degree $\leq 4$, we may assume without loss of generality that the minimal counterexample $G$ has minimum degree 5. We now show that $G$ must contain at least 12 vertices of degree exactly 5. + +\begin{lemma} +If $G$ is a planar graph with minimum degree 5, then $G$ contains at least 12 vertices of degree exactly 5. +\end{lemma} + +\begin{proof} +For each $k \geq 5$, let $n_k$ denote the number of vertices of degree exactly $k$ in $G$. Since every vertex has degree $\geq 5$, +\[ + V = \sum_{k \geq 5} n_k, \qquad 2E = \sum_{k \geq 5} k\, n_k. +\] +Using $E \leq 3V - 6$, we obtain $2E \leq 6V - 12$, so +\[ + \sum_{k \geq 5} k\, n_k \leq 6\sum_{k \geq 5} n_k - 12. +\] +Moving all terms to the right-hand side gives +\[ + 12 \leq 6\sum_{k \geq 5} n_k - \sum_{k \geq 5} k\, n_k = \sum_{k \geq 5} (6 - k)\, n_k. +\] +We now expand this sum by separating the $k = 5$, $k = 6$, and $k \geq 7$ terms: +\[ + \sum_{k \geq 5}(6-k)\,n_k + = \underbrace{(6-5)\,n_5}_{=\,n_5} + + \underbrace{(6-6)\,n_6}_{=\,0} + + \sum_{k \geq 7}(6-k)\,n_k + = n_5 - \sum_{k \geq 7}(k-6)\,n_k. +\] +Therefore $12 \leq n_5 - \sum_{k \geq 7}(k-6)\,n_k$, which rearranges to +\[ + n_5 \geq 12 + \sum_{k \geq 7}(k-6)\,n_k. +\] +Since $k - 6 \geq 1 > 0$ and $n_k \geq 0$ for all $k \geq 7$, each term in the sum is non-negative, so $\sum_{k \geq 7}(k-6)\,n_k \geq 0$ and thus $n_5 \geq 12$. +\end{proof} + +\subsection{Two Non-Adjacent Vertices of Degree 5} + +\begin{lemma} +If $G$ is a planar graph with minimum degree 5, then $G$ contains two non-adjacent vertices of degree exactly 5. +\end{lemma} + +\begin{proof} +By the previous lemma, $G$ contains at least 12 vertices of degree exactly 5. Let $S$ denote the set of all degree-5 vertices, so $|S| \geq 12$. Suppose for contradiction that every two vertices in $S$ are adjacent, i.e., $S$ induces a clique in $G$. Then every vertex $v \in S$ is adjacent to all other $|S| - 1 \geq 11$ vertices of $S$. But $\deg(v) = 5$, so $v$ has at most 5 neighbors in total. Since $11 > 5$, this is a contradiction. Therefore $S$ does not form a clique, and there exist two vertices $v_1, v_2 \in S$ that are non-adjacent. +\end{proof} + +\subsection{The Reduced Subgraph $G'$} + +By the previous lemma, we may fix two non-adjacent vertices $v_0, v_1 \in G$ each of degree 5. Define +\[ + G' = G - \{v_0, v_1\}, +\] +as the subgraph of $G$ obtained by deleting $v_0$ and $v_1$ together with all edges incident to either vertex. Since $G'$ has strictly fewer vertices than $G$, the minimality of $G$ guarantees that $G'$ admits a proper 4-coloring $\phi : V(G') \to \{1,2,3,4\}$. Because $v_0$ and $v_1$ are non-adjacent in $G$, neither is a neighbor of the other, so each retains all 5 of its neighbors in $G'$. Let $N(v_i)$ denote the set of neighbors of $v_i$ in $G$; then $N(v_0)$ and $N(v_1)$ are each sets of 5 vertices, all of which lie in $G'$ and are assigned colors by $\phi$. Note also that $N(v_0)$ and $N(v_1)$ may overlap. Let $\Phi$ be the set of all possible 4-colorings of $G'$. + +\subsection{Not All Colorings in $\Phi$ Saturate Both Neighborhoods} + +\begin{lemma} +It is not possible that every $\phi \in \Phi$ satisfies both $|\phi(N(v_0))| = 4$ and $|\phi(N(v_1))| = 4$. +\end{lemma} + +\begin{proof} +We show the stronger fact that there exists $\phi \in \Phi$ with $|\phi(N(v_0))| \leq 3$. + +Since $v_1$ is a single vertex, $G - \{v_1\}$ is a planar graph on $V - 1$ vertices. Because $V - 1 < V$ and $G$ is a minimal counterexample, $G - \{v_1\}$ is 4-colorable. Fix a proper 4-coloring $\psi$ of $G - \{v_1\}$. This coloring assigns a color to every vertex of $G'$ and also to $v_0$. + +Let $\phi = \psi|_{V(G')}$ be the restriction of $\psi$ to $G'$. Since $\psi$ is a proper coloring of $G - \{v_1\} \supseteq G'$, the restriction $\phi$ is a proper 4-coloring of $G'$, so $\phi \in \Phi$. + +Because $\psi$ is a proper coloring of $G - \{v_1\}$ and $v_0 \in V(G - \{v_1\})$, the color $\psi(v_0)$ differs from the color of every neighbor of $v_0$. In particular, $\psi(v_0) \notin \phi(N(v_0))$. Since $\psi(v_0) \in \{1,2,3,4\}$ and $\psi(v_0)$ does not appear in $\phi(N(v_0))$, it follows that $|\phi(N(v_0))| \leq 3$. Therefore not every coloring in $\Phi$ uses all 4 colors on $N(v_0)$, and in particular it is impossible that every $\phi \in \Phi$ saturates both $N(v_0)$ and $N(v_1)$ with all 4 colors. +\end{proof} + +\section{Merged Subgraphs of $G'$} + +Let $N(v_0) = \{a_0, a_1, a_2, a_3, a_4\}$ be the five neighbors of $v_0$ in $G'$. For each pair of non-adjacent vertices $a_i, a_j \in N(v_0)$ (i.e., $\{a_i, a_j\} \notin E(G')$), define the graph $G'_{ij}$ to be the result of identifying $a_i$ and $a_j$ in $G'$: formally, $G'_{ij}$ is obtained from $G'$ by removing $a_i$ and $a_j$ and adding a new vertex $a_{ij}$ adjacent to every vertex in $N_{G'}(a_i) \cup N_{G'}(a_j)$. Define the set of all such merged subgraphs by +\[ + \mathcal{M} = \bigl\{\, G'_{ij} : a_i, a_j \in N(v_0),\ \{a_i, a_j\} \notin E(G') \,\bigr\}. +\] + +\subsection{Colorings of Merged Subgraphs Extend to $G'$} + +For each $G'_{ij} \in \mathcal{M}$, let $\Phi(G'_{ij})$ denote the set of proper 4-colorings of $G'_{ij}$. Each such coloring can be extended to a coloring of $G'$ by assigning the color of the merged vertex $a_{ij}$ back to both $a_i$ and $a_j$. Formally, for $\psi \in \Phi(G'_{ij})$, define $\widetilde{\psi} : V(G') \to \{1,2,3,4\}$ by +\[ + \widetilde{\psi}(v) = \begin{cases} \psi(a_{ij}) & \text{if } v = a_i \text{ or } v = a_j, \\ \psi(v) & \text{otherwise.} \end{cases} +\] +Let $\widetilde{\Phi}_{ij} = \{\,\widetilde{\psi} : \psi \in \Phi(G'_{ij})\,\}$ denote the set of all such extensions. + +\begin{lemma} +For each $G'_{ij} \in \mathcal{M}$, we have $\widetilde{\Phi}_{ij} \subseteq \Phi$. +\end{lemma} + +\begin{proof} +Let $\psi \in \Phi(G'_{ij})$ and let $\widetilde{\psi}$ be its extension to $G'$. We verify that $\widetilde{\psi}$ is a proper coloring of $G'$ by checking every edge $\{u, w\} \in E(G')$. + +\textbf{Case 1: neither $u$ nor $w$ is $a_i$ or $a_j$.} Then $\{u, w\}$ is also an edge of $G'_{ij}$, and $\widetilde{\psi}(u) = \psi(u) \neq \psi(w) = \widetilde{\psi}(w)$. + +\textbf{Case 2: $u \in \{a_i, a_j\}$ and $w \notin \{a_i, a_j\}$ (or vice versa).} Then $w \in N_{G'}(a_i) \cup N_{G'}(a_j)$, so $w$ is adjacent to $a_{ij}$ in $G'_{ij}$. Since $\psi$ is proper, $\psi(w) \neq \psi(a_{ij}) = \widetilde{\psi}(u)$. + +\textbf{Case 3: $u = a_i$ and $w = a_j$.} This case cannot occur, since $a_i$ and $a_j$ are non-adjacent in $G'$ by the definition of $\mathcal{M}$. + +In all cases the endpoints of every edge receive distinct colors, so $\widetilde{\psi} \in \Phi$. +\end{proof} + +\subsection{4-Saturating Merged Colorings Extend to 4-Saturating Colorings of $G'$} + +For each $G'_{ij} \in \mathcal{M}$, the vertices of $G'_{ij}$ that would be adjacent to $v_0$ if $v_0$ were reinserted are exactly $\{a_{ij}\} \cup (N(v_0) \setminus \{a_i, a_j\})$; call this set $N_{ij}(v_0)$. Define +\[ + \Phi_4 = \{\, \phi \in \Phi : |\phi(N(v_0))| = 4 \,\} +\] +to be the set of colorings of $G'$ that use all 4 colors on $N(v_0)$, and for each $G'_{ij} \in \mathcal{M}$ define +\[ + \Phi_4(G'_{ij}) = \{\, \psi \in \Phi(G'_{ij}) : |\psi(N_{ij}(v_0))| = 4 \,\} +\] +to be the colorings of $G'_{ij}$ that use all 4 colors on the effective neighborhood of $v_0$. + +\begin{lemma} +For each $G'_{ij} \in \mathcal{M}$, the extensions of $\Phi_4(G'_{ij})$ lie in $\Phi_4$, i.e., $\widetilde{\Phi_4(G'_{ij})} \subseteq \Phi_4$. +\end{lemma} + +\begin{proof} +Let $\psi \in \Phi_4(G'_{ij})$, so $|\psi(N_{ij}(v_0))| = 4$. By definition of the extension, +\[ + \widetilde{\psi}(N(v_0)) + = \widetilde{\psi}(\{a_i, a_j\} \cup (N(v_0) \setminus \{a_i, a_j\})) + = \{\psi(a_{ij})\} \cup \psi(N(v_0) \setminus \{a_i, a_j\}) + = \psi(N_{ij}(v_0)). +\] +The first equality expands $N(v_0)$; the second uses $\widetilde{\psi}(a_i) = \widetilde{\psi}(a_j) = \psi(a_{ij})$ and $\widetilde{\psi}(u) = \psi(u)$ for $u \notin \{a_i, a_j\}$; the third uses $N_{ij}(v_0) = \{a_{ij}\} \cup (N(v_0) \setminus \{a_i,a_j\})$. Therefore $|\widetilde{\psi}(N(v_0))| = |\psi(N_{ij}(v_0))| = 4$. Since also $\widetilde{\psi} \in \Phi$ by the previous lemma, we conclude $\widetilde{\psi} \in \Phi_4$. +\end{proof} + \end{document} %-----------------------------------------------------------------------