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remove kempe_proof_fixed and kempe_style_search_for_smaller_contradiction

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Both papers reproduced the classical (and famously incomplete) 1879
Kempe argument: degree ≤ 4 case via Kempe chains is well known, and
both stopped at the unresolved degree-5 case (exactly where Kempe
stopped). They were near-duplicates of each other, contained no novel
mathematical claim, and were not currently publishable.

Removing to keep the papers/ directory focused on active or
self-contained work.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-25 13:46:14 -04:00
parent 74030a5b8f
commit 04dd72076b
14 changed files with 0 additions and 2112 deletions
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papers/kempe_proof_fixed/paper.aux
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\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{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 }
\@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 }
\@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 }
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{0pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{29.38873pt}
\newlabel{tocindent3}{0pt}
\@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 }
\@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 }
\gdef \@abspage@last{5}
papers/kempe_proof_fixed/paper.fdb_latexmk
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%% filename: amsart-template.tex
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%% American Mathematical Society
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%% Copyright 2008-2010, 2014 American Mathematical Society.
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%% of this license or (at your option) any later version.
%% The latest version of this license is in
%% http://www.latex-project.org/lppl.txt
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%% version 2005/12/01 or later.
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%% This work has the LPPL maintenance status `maintained'.
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%% The Current Maintainer of this work is the American Mathematical
%% Society.
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%% ====================================================================
% AMS-LaTeX v.2 template for use with amsart
%
% Remove any commented or uncommented macros you do not use.
\documentclass{amsart}
\usepackage{tikz}
\usetikzlibrary{hobby}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\begin{document}
\title{Humans Suffice: A Novel Proof of the Four Color Theorem}
% Remove any unused author tags.
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{eric@dideric.is}
\thanks{Dr. Joe Fields}
\thanks{Holepunch}
% author two information
% \author{}
% \address{}
% \curraddr{}
% \email{}
% \thanks{}
\subjclass[2020]{Primary }
\keywords{Discrete , Four Color Theorem,}
\date{}
\dedicatory{Dedicated to those who doubt the machine}
\begin{abstract}
\end{abstract}
\maketitle
\section{Kempe's Proof (Valid Portion)}
\subsection{Setup}
Kempe's strategy, published in 1879, 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}
\begin{figure}[h]
\centering
\begin{tikzpicture}[every node/.style={circle, draw, fill=black, inner sep=2pt, minimum size=5pt}]
\node[label=below:{$v_0$}] (v0) at (0,0) {};
\node[label=above:{$a_0$}] (a0) at (90:1.8) {};
\node[label=left:{$a_1$}] (a1) at (162:1.8) {};
\node[label=left:{$a_2$}] (a2) at (234:1.8) {};
\node[label=right:{$a_3$}] (a3) at (306:1.8) {};
\node[label=right:{$v_1$}] (v1) at (18:1.8) {};
\draw (a0) -- (a1) -- (a2) -- (a3) -- (v1) -- (a0);
\draw (v0) -- (a0);
\draw (v0) -- (a1);
\draw (v0) -- (a2);
\draw (v0) -- (a3);
\draw (v0) -- (v1);
\draw[dotted] (a0) -- ++(75:0.7) (a0) -- ++(105:0.7);
\draw[dotted] (a1) -- ++(147:0.7) (a1) -- ++(177:0.7);
\draw[dotted] (a2) -- ++(219:0.7) (a2) -- ++(249:0.7);
\draw[dotted] (a3) -- ++(291:0.7) (a3) -- ++(321:0.7);
\draw[dotted] (v1) -- ++(3:0.7) (v1) -- ++(33:0.7);
\end{tikzpicture}
\caption{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.}
\end{figure}
Let $G$ be a minimum example of a maximal planar graph requiring 5 colors, and let $v_0$ be a graph in $G$ of degree 5, and let $v_1$ be a neighbor of $v_0$.
\begin{lemma}
Let $G^*$ be the subgraph of $G$ obtained by deleting the edge $\{v_0, v_1\}$, and let $G'$ be the graph obtained by merging vertices $a_0$ and $a_3$ in $G^*$. Then there is at least one proper 4-coloring of $G*$ which assigns $v_0$ and $v_1$ the same color and $a_0$ and $a_3$ the same color.
\end{lemma}
\begin{proof}
Since $G'$ has fewer vertices than $G$, it admits a proper 4-coloring $\phi$ by minimality. Any such $\phi$ must assign $v_0$ and $v_1$ the same color: if they received distinct colors, $\phi$ would induce a proper 4-coloring of $G^*$ in which $v_0 \neq v_1$, and re-inserting edge $\{v_0,v_1\}$ would yield a proper 4-coloring of $G$, contradicting minimality. The induced coloring $\Phi$ of $G^*$ assigns $a_0$ and $a_3$ the common color $\phi(a_{03})$ by construction, so $\Phi$ is the required coloring.
\end{proof}
\begin{figure}[h]
\centering
\begin{tikzpicture}[every node/.style={circle, draw, fill=black, inner sep=2pt, minimum size=5pt}]
\node[label=below:{$v_0$}] (v0) at (0,0) {};
\node[label=above:{$a_0$}] (a0) at (90:1.8) {};
\node[label=left:{$a_1$}] (a1) at (162:1.8) {};
\node[label=left:{$a_2$}] (a2) at (234:1.8) {};
\node[label=right:{$a_3$}] (a3) at (306:1.8) {};
\node[label=right:{$v_1$}] (v1) at (18:1.8) {};
\draw (a0) -- (a1) -- (a2) -- (a3) -- (v1) -- (a0);
\draw (v0) -- (a0);
\draw (v0) -- (a1);
\draw (v0) -- (a2);
\draw (v0) -- (a3);
\draw[dotted] (a0) -- ++(75:0.7) (a0) -- ++(105:0.7);
\draw[dotted] (a1) -- ++(147:0.7) (a1) -- ++(177:0.7);
\draw[dotted] (a2) -- ++(219:0.7) (a2) -- ++(249:0.7);
\draw[dotted] (a3) -- ++(291:0.7) (a3) -- ++(321:0.7);
\draw[dotted] (v1) -- ++(3:0.7) (v1) -- ++(33:0.7);
\end{tikzpicture}
\caption{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.}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}[every node/.style={circle, draw, fill=black, inner sep=2pt, minimum size=5pt}]
\node[label=below:{$v_0$}] (v0) at (0,0) {};
\node[label=above:{$a_{03}$}] (a03) at (18:0.9) {};
\node[label=left:{$a_1$}] (a1) at (162:1.8) {};
\node[label=left:{$a_2$}] (a2) at (234:1.8) {};
\node[label=right:{$v_1$}] (v1) at (18:1.8) {};
\draw (a03) -- (a1) -- (a2) -- (a03);
\draw (a03) -- (v1);
\draw (v0) -- (a03);
\draw (v0) -- (a1);
\draw (v0) -- (a2);
\draw[dotted] (a03) -- ++(75:0.7) (a03) -- ++(105:0.7);
\draw[dotted] (a03) -- ++(255:0.7) (a03) -- ++(285:0.7);
\draw[dotted] (a1) -- ++(147:0.7) (a1) -- ++(177:0.7);
\draw[dotted] (a2) -- ++(219:0.7) (a2) -- ++(249:0.7);
\draw[dotted] (v1) -- ++(3:0.7) (v1) -- ++(33:0.7);
\end{tikzpicture}
\caption{The neighborhood of $v_0$ and $v_1$ in $G'$: $G^*$ with $a_0$ and $a_3$ merged into $a_{03}$.}
\end{figure}
\begin{lemma}
Let $\Phi$ be a proper 4-coloring of $G*$ induced by a proper 4-coloring of $G'$, let $\alpha = \Phi(v_0) = \Phi(v_1)$, and let $\beta = \Phi(a_{0}) = \Phi(a_{3})$. Then for each of the two remaining colors $\gamma$ and $\delta$, there exists a Kempe chain in $G'$ connecting $v_0$ to $v_1$ using colors $\{\alpha, \gamma\}$ and $\{\alpha, \delta\}$ respectively.
\end{lemma}
\begin{proof}
We show $v_0$ and $v_1$ lie in the same $\{\alpha, \gamma\}$-Kempe chain; the argument for $\{\alpha, \delta\}$ is identical.
By the previous lemma, $\Phi$ must exist. Suppose for contradiction that $v_0$ and $v_1$ lie in \emph{different} $\{\alpha, \gamma\}$-Kempe chains. Swapping colors $\alpha$ and $\gamma$ throughout $v_0$'s chain yields a new proper 4-coloring $\Phi'$ of $G*$ in which $\Phi'(v_0) = \gamma$ while $\Phi'(v_1) = \alpha$. Since $\gamma \neq \alpha$, the coloring $\Phi'$ assigns $v_0$ and $v_1$ distinct colors, which allows us to properly color $G$. Therefore $v_0$ and $v_1$ lie in the same $\{\alpha, \gamma\}$-Kempe chain, and by the same argument they lie in the same $\{\alpha, \delta\}$-Kempe chain.
\end{proof}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\coordinate (v0pos) at (0,0);
\coordinate (a03pos) at (18:0.9);
\coordinate (a1pos) at (162:1.8);
\coordinate (a2pos) at (234:1.8);
\coordinate (v1pos) at (18:1.8);
% Kempe chain highlights beneath nodes
\draw[line width=4pt, green!50, dashed] (a1pos) to[bend left=40] (v1pos);
\draw[line width=4pt, green!80, dashed] (a1pos) to (v0pos);
\draw[line width=4pt, yellow!80, dashed] (a2pos) to[bend right=40] (v1pos);
\draw[line width=4pt, yellow!80, dashed] (a2pos) to (v0pos);
% Regular edges
\draw (a03pos) -- (a1pos) -- (a2pos) -- (a03pos);
\draw (a03pos) -- (v1pos);
\draw (v0pos) -- (a03pos);
\draw (v0pos) -- (a1pos);
\draw (v0pos) -- (a2pos);
% Dotted lines
\draw[dotted] (a03pos) -- ++(75:0.7) (a03pos) -- ++(105:0.7);
\draw[dotted] (a03pos) -- ++(255:0.7) (a03pos) -- ++(285:0.7);
\draw[dotted] (a1pos) -- ++(147:0.7) (a1pos) -- ++(177:0.7);
\draw[dotted] (a2pos) -- ++(219:0.7) (a2pos) -- ++(249:0.7);
\draw[dotted] (v1pos) -- ++(3:0.7) (v1pos) -- ++(33:0.7);
% Colored nodes
\node[circle, draw, fill=red!60, inner sep=2pt, minimum size=10pt, label=below:{$v_0$}] at (v0pos) {};
\node[circle, draw, fill=blue!60, inner sep=2pt, minimum size=10pt, label=above:{$a_{03}$}] at (a03pos) {};
\node[circle, draw, fill=green!60, inner sep=2pt, minimum size=10pt, label=left:{$a_1$}] at (a1pos) {};
\node[circle, draw, fill=yellow!80, inner sep=2pt, minimum size=10pt, label=left:{$a_2$}] at (a2pos) {};
\node[circle, draw, fill=red!60, inner sep=2pt, minimum size=10pt, label=right:{$v_1$}] at (v1pos) {};
\end{tikzpicture}
\caption{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 $a_2$).}
\end{figure}
\begin{figure}[h]
\centering
\begin{tikzpicture}[every node/.style={circle, draw, inner sep=2pt, minimum size=10pt}]
\coordinate (v0pos) at (0,0);
\coordinate (a0pos) at (90:1.8);
\coordinate (a1pos) at (162:1.8);
\coordinate (a2pos) at (234:1.8);
\coordinate (a3pos) at (306:1.8);
\coordinate (v1pos) at (18:1.8);
\coordinate (green_mid) at (0, 2.4);
\coordinate (yellow_mid) at (306:2.3);
% Kempe chain highlights beneath nodes
\draw[line width=4pt, green!50, dashed] (a1pos) to[curve through={(green_mid)}] (v1pos);
\draw[line width=4pt, green!80, dashed] (a1pos) to (v0pos);
\draw[line width=4pt, yellow!80, dashed] (a2pos) to[curve through={(yellow_mid)}] (v1pos);
\draw[line width=4pt, yellow!80, dashed] (a2pos) to (v0pos);
\node[fill=red!60, label=below:{$v_0$}] (v0) at (v0pos) {};
\node[fill=blue!60, label=above:{$a_0$}] (a0) at (a0pos) {};
\node[fill=green!60, label=left:{$a_1$}] (a1) at (a1pos) {};
\node[fill=yellow!80, label=left:{$a_2$}] (a2) at (a2pos) {};
\node[fill=blue!60, label=right:{$a_3$}] (a3) at (a3pos) {};
\node[fill=red!60, label=right:{$v_1$}] (v1) at (v1pos) {};
\draw (a0) -- (a1) -- (a2) -- (a3) -- (v1) -- (a0);
\draw (v0) -- (a0);
\draw (v0) -- (a1);
\draw (v0) -- (a2);
\draw (v0) -- (a3);
\draw[dotted] (a0) -- ++(75:0.7) (a0) -- ++(105:0.7);
\draw[dotted] (a1) -- ++(147:0.7) (a1) -- ++(177:0.7);
\draw[dotted] (a2) -- ++(219:0.7) (a2) -- ++(249:0.7);
\draw[dotted] (a3) -- ++(291:0.7) (a3) -- ++(321:0.7);
\draw[dotted] (v1) -- ++(3:0.7) (v1) -- ++(33:0.7);
\end{tikzpicture}
\caption{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.}
\end{figure}
\begin{lemma}
Let $\Phi$ be a proper 4-coloring of $G*$ induced by a proper 4-coloring of $G'$, let $\alpha = \Phi(v_0) = \Phi(v_1)$, let $\beta = \Phi(a_{0}) = \Phi(a_{3})$, let $\delta = \Phi(a_{2})$, and let $\gamma = \Phi(a_{1})$. Then the coloring $\Phi'$ obtained by doing a Kempe switch on the $\{\beta, \delta\}$-Kempe chain incident to $a_0$ must assign $\alpha = \Phi'(v_0) = \Phi'(v_1)$, $\beta = \Phi'(a_3)$, $\delta = \Phi'(a_0) = \Phi'(a_2)$, and $\gamma = \Phi'(a_1)$ (IE the coloring for all vertices except $a_0$ remains the same)
\end{lemma}
\begin{proof}
The kempe switch will assign $\Phi'(a_0) = \delta$ by definition. By the previous lemma, there must be a $\{\alpha, \gamma\}$ kempe chain incident to $v_0$, $v_1$, and $a_1$. Therefore the $\{\beta, \delta\}$ component incident to $a_2$ and $a_3$ cannot be connected to the $\{\beta, \delta\}$ component incident to $a_0$, so the kempe switch we do to obtain $\Phi'$ will not recolor vertices $a_2$ and $a_3$. Since the $\{\beta, \delta\}$ kempe switch also maintains vertices assigned $\alpha$ and $\gamma$ in $\Phi$, $\Phi'$ will not recolor vertices $a_1$, $v_0$, and $v_1$. Therefore $\alpha = \Phi'(v_0) = \Phi'(v_1)$, $\beta = \Phi'(a_3)$, $\delta = \Phi'(a_0) = \Phi'(a_2)$, and $\gamma = \Phi'(a_1)$.
\end{proof}
\begin{figure}[h]
\centering
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\begin{document}
\title{Humans Suffice: A Novel Proof of the Four Color Theorem}
% Remove any unused author tags.
% author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{eric@dideric.is}
\thanks{Dr. Joe Fields}
\thanks{Holepunch}
% author two information
% \author{}
% \address{}
% \curraddr{}
% \email{}
% \thanks{}
\subjclass[2020]{Primary }
\keywords{Discrete , Four Color Theorem,}
\date{}
\dedicatory{Dedicated to all who value more than machines}
\begin{abstract}
\end{abstract}
\maketitle
\section{Kempe's Proof (Valid Portion)}
\subsection{Setup}
Kempe's strategy, published in 1879, 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}
\subsection{Locked Colorings}
\begin{definition}
Let $a_i, a_j \in N(v_0)$ be non-adjacent in $G'$ (where $G' = G - \{v_0, v_1\}$ as in Section 2.1). A coloring $\phi \in \Phi$ is \emph{locked relative to $\{a_i, a_j\}$} if
\[
|\phi(N_{G'}(a_i))| > 2 \quad \text{or} \quad |\phi(N_{G'}(a_j))| > 2.
\]
Denote the set of all such colorings by $\Lambda_{ij} \subseteq \Phi$.
\end{definition}
Intuitively, a coloring is locked relative to $\{a_i, a_j\}$ when the neighborhood of at least one of the two vertices is colored with enough distinct colors to obstruct a Kempe chain swap that would free a color for $v_0$.
\begin{lemma}
Let $G'_{ij} \in \mathcal{M}$ and let $\psi \in \Phi(G'_{ij})$. If $|\psi(N_{G'_{ij}}(v_1))| \leq 3$, then $\psi$ is the induced coloring of some locked coloring in $\Phi$.
\end{lemma}
\begin{proof}
The extension $\widetilde{\psi} \in \Phi$ (by the lemma in Section 3.1). Suppose for contradiction that $\widetilde{\psi} \notin \Lambda_{ij}$, i.e., $\widetilde{\psi}$ is not locked relative to $\{a_i, a_j\}$. Then
\[
|\widetilde{\psi}(N_{G'}(a_i))| \leq 2 \quad \text{and} \quad |\widetilde{\psi}(N_{G'}(a_j))| \leq 2.
\]
Since $\widetilde{\psi}$ agrees with $\psi$ on all vertices of $G'_{ij}$, the neighborhoods of $a_i$ and $a_j$ in $G'$ each use at most 2 colors under $\widetilde{\psi}$. With so few colors in each neighborhood, a Kempe chain swap can be performed in $\widetilde{\psi}$ to give $a_i$ and $a_j$ the same color, freeing a fourth color for $v_0$. Simultaneously, since $|\psi(N_{G'_{ij}}(v_1))| \leq 3$, the vertex $v_1$ can be assigned the remaining color not used by its neighbors. Together these assignments extend $\widetilde{\psi}$ to a proper 4-coloring of all of $G$, contradicting the minimality of $G$ as a counterexample. Therefore $\widetilde{\psi} \in \Lambda_{ij}$, and $\psi$ is the induced coloring of the locked coloring $\widetilde{\psi}$.
\end{proof}
\begin{lemma}
Now prove that all colorings of every merged graph relative to $\{a_i, a_j\}$ must be a locked coloring.
\end{lemma}
\begin{proof}
\end{proof}
\begin{lemma}
If all $\phi \in \Phi$ are locked colorings with respect to all pairs of non adjacent vertices $\{a_i, a_j\} \in N(v_0)$, then all colorings of all merged graphs with respect to ${a_k, a_l} \in N(v_1)$ require 4 colors for $N_(v_0)$.
\end{lemma}
\begin{proof}
\end{proof}
\end{document}
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