\documentclass[11pt]{article} \usepackage[margin=1in,headheight=24pt]{geometry} \usepackage{fancyhdr} \setlength{\headheight}{55pt} \usepackage{hyperref} \usepackage{tcolorbox} \usepackage{xcolor} \usepackage{amsfonts,amsmath,amssymb,amsthm} \usepackage{mathtools} \usepackage{subcaption} \usepackage{tikz} \usepackage{tikz-network} \newtheorem{theorem}{Theorem}[section] \newtheorem{axiom}[theorem]{Axiom} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{fact}[theorem]{Fact} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \definecolor{black}{RGB}{0,0,0} \definecolor{orange}{RGB}{230,159,0} \definecolor{skyblue}{RGB}{86,180,233} \definecolor{bluishgreen}{RGB}{0,158,115} \definecolor{yellow}{RGB}{240,228,66} \definecolor{blue}{RGB}{0,114,178} \definecolor{vermillion}{RGB}{213,94,0} \definecolor{reddishpurple}{RGB}{204,121,167} \definecolor{cugold}{RGB}{207,184,124} \pagestyle{plain} \fancypagestyle{firstpage}{ \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[c]{ \makebox[\textwidth][l]{\textbf{MATH 6404: Applied [Combinatorics and] Graph Theory} \hfill CU Denver} \\ \rule{\textwidth}{0.5pt} \\ \makebox[\textwidth][l]{Spring 2026 \hfill Instructor: Carlos Mart\'inez} } \fancyfoot[C]{\thepage} } \newcommand{\scribebox}[4]{ \begin{tcolorbox}[colback=cugold!40,colframe=black,left=6pt,right=6pt,top=10pt,bottom=10pt] \centering \textbf{Lecture #1:} #2 \\ \textbf{Date:} #3 \hfill \textbf{Scribe:} #4 \end{tcolorbox} } %%% -+-+-+-+-+-+- BEGIN HERE -+-+-+-+-+-+- %%% \newcommand{\lecturenumber}{$21$} \newcommand{\lecturetitle}{Coloring Planar Graphs} %\newcommand{\scribename}{Carlos Mart\'inez} \newcommand{\scribename}{Mark Johnson} \newcommand{\lecturedate}{April 13, 2026} \begin{document} \thispagestyle{firstpage} \scribebox{\lecturenumber}{\lecturetitle}{\lecturedate}{\scribename} \section{Greedy Coloring} For a graph $G_1$, the \textbf{greedy coloring} w.r.t. an ordering $v_1, v_2,..., v_n$ of its nodes, colors each node in that order and gives $v_i$ the least index color not used by its earlier-colored neighbors. \begin{itemize} \item This need not produce an optimal coloring, but whatever coloring it produces gives an upper-bound to the $\chi(G)$. \\ \end{itemize} \textbf{Proposition:} $\chi(G) \le \triangle(G)+1$, where $\triangle(G)$ is the maximum degree in $G$. \\ \begin{tcolorbox}[colback=cugold!40,colframe=black,left=6pt,right=6pt,top=10pt,bottom=10pt] \textbf{Proof} \\ In any ordering of $V(G)$, each node has at most $\triangle(G)$ earlier-colored neighbors. This implies that the Greedy Algorithm will use no more than $\triangle(G)+1$ colors. \\ \qed \end{tcolorbox} A graph is \textbf{$k$-degenerate} if every subgraph has a node of degree at most $k$. \\ A \textbf{smallest-last ordering} of $V$ is constructed iteratively by letting $v_i$ be a node of minimum degree in $G-\{v_1, v_2, ..., v_{i-1}\}$ \\ \textbf{Proposition (Szkeras-Wilf):} If $G$ is $k$-degenerate, then $G$ is $(k+1)$-colorable. \\ \begin{tcolorbox}[colback=cugold!40,colframe=black,left=6pt,right=6pt,top=10pt,bottom=10pt] \textbf{Proof} \\ If $G$ is $k$-degenerate, then the smallest-last ordering gives each node at most $k$ neighbors among the earlier-colored nodes. This implies that the greedy coloring w.r.t. this ordering used at most $k+1$ colors.\\ \qed \end{tcolorbox} \newpage \section{How Does This Relate to Planar Graphs?} From lecture 20, we know that a planar graph with $n$ nodes and $m$ edges satisfies $m\le 3n-6$ (this followed from Euler's formula). Therefore, planar graphs are $5$-degenerate. $$\sum_{u\in V} \text{deg}(u)=2m\le 6n-12$$ $$\frac{1}{n}\sum_{u\in V} \text{deg}(u)\le \frac{6n-12}{n}=6-\frac{12}{n}<6$$ $\implies$ the average node has degree $<6$, but degree is an integer quantity. \\ $\implies$ for every planar graph, there must exist a node of degree $\le 5$. \\ Together with our proposition (Szkeras-Wilf), we find that planar graphs are $6$-colorable. \section{A Stronger Result} \subsection{Setup} A graph $G$ is \textbf{color-critical} if $X(H)<\chi(G)$ for all subgraphs $H\subseteq G$. If it also holds that $\chi(G)=K$, we say that $G$ is $K$-critical. \\ \textbf{Proposition} If $G$ is $K$-critical, then $\delta(G)\ge K-1$, where $\delta(G)$ is the minimum degree of $G$. \\ \begin{tcolorbox}[colback=cugold!40,colframe=black,left=6pt,right=6pt,top=10pt,bottom=10pt] \textbf{Proof} \\ If $G$ is $K$-critical, then $\chi(G-u)=K-1$ for all $u\in V$. \\ Now consider any $u\in V$. Since $\chi(G-u)=K-1$, there exists a proper coloring of $G-u$ with exactly $K-1$ colors (by definition of $\chi$). \\ If deg$(u)