\documentclass[11pt]{article}
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%% header definitions
\def\course{MATH 314}
\def\assign{HW \#11}
\def\due{Apr 19}
\def\doclink{\url{https://mathematicaster.org/teaching/graphs2022/hw/hw11.tex}}

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\newtheorem{problem}{Problem}

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\begin{document}
\pagestyle{empty}

\noindent
\begin{minipage}{.33\textwidth}
	\flushleft {\bf \course}
\end{minipage}
\begin{minipage}{.33\textwidth}
	\centering {\bf \assign}
\end{minipage}
\begin{minipage}{.33\textwidth}
	\flushright {\bf \due}
\end{minipage}
\vskip3pt
\hrule\hrule\hrule
\vskip5pt

\noindent {\footnotesize This document is from \doclink}
\vskip10pt

\noindent {\small Unless explicitly requested by a problem, do not include sketches as part of your proof.
You are free to use the result from any problem on this (or previous) assignment as a part of your solution to a different problem even if you have not solved the former problem.
}

\vskip20pt

\begin{problem}[2pts]
    Let $G$ be any graph.
    Prove that
    \[
        \abs{E(G)}\geq{\chi(G)\choose 2}.
    \]
\end{problem}




\begin{problem}[1pt]
    Generalize Theorem 1 from 04-07:

    Let $G=(V,E)$ be any graph and consider coloring the edges of $G$ red and blue (formally, we have a function $f\colon E\to\{\text{red},\text{blue}\}$).
    Let $G_r$ be the graph formed by the red edges and let $G_b$ be the graph formed by the blue edges (formally, $G_r=\bigl(V,f^{-1}(\text{red})\bigr)$ and $G_b=\bigl(V,f^{-1}(\text{blue})\bigr)$).
    Prove that
    \[
        \chi(G_r)\cdot\chi(G_b)\geq\chi(G).
    \]
\end{problem}



\begin{problem}[1pt]
    For every pair of positive integers $m,n$, construct a graph $G$ with the following properties:
    \begin{itemize}
        \item $G$ has $m\cdot n$ many vertices, and
        \item $\chi(G)=m$, and
        \item $\chi(\wbar G)=n$.
    \end{itemize}
\end{problem}



\begin{problem}[2pts]
    Prove that $G$ is $3$-critical if and only if $G\cong C_{2n+1}$ for some positive integer $n$.
\end{problem}


\begin{problem}[2pts]\label{neighborchi}
    Fix any integer $k\geq 1$.
    Prove that if $G$ is an $n$-vertex graph wherein
    \[
        \chi\bigl(G[N(v)]\bigr)\leq k,\quad\text{for every }v\in V(G),
    \]
    then $\chi(G)\leq\sqrt{2kn}$.

    (Hint: Take motivation from our proof that $\chi(G)\leq\sqrt{2n}$ if $G$ is triangle-free. Be warned, though, there are a couple steps which require more care.)

    (Hint: If $0\leq x\leq y$, then $x\leq\sqrt{xy}$.)
\end{problem}


\begin{problem}[2pts]
    For graphs $G,H$, the graph $G$ is said to be $H$-free if $G$ does not contain a copy of $H$.

    Fix any integer $k\geq 3$.
    Prove that if $G$ is a $C_k$-free graph on $n$ vertices, then $\chi(G)\leq\sqrt{2(k-2)n}$.

    (Hint: Problem~\ref{neighborchi})
\end{problem}




\end{document}
