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-\documentclass[11pt]{article}
-\usepackage[margin=1in]{geometry}
-\usepackage{amsfonts,amsmath,amssymb,amsthm}
-
-\PassOptionsToPackage{obeyspaces}{url}
-\usepackage[colorlinks=true,citecolor=blue,urlcolor=blue,linkcolor=blue,bookmarksopen=true]{hyperref}
-\usepackage{tikz}
-
-\usepackage{enumitem}
-\setlist{itemsep=0.5em}
-
-%% header definitions
-\def\course{MATH 314}
-\def\courselink{https://mathematicaster.org/teaching/graphs2022}
-\def\assign{HW \#4}
-\def\assignlink{hw4.tex}
-\def\due{Feb 8}
-\def\doctype{This homework is from}
-
-%% environments
-% theorem style
-\newtheorem{theorem}{Theorem}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-\newtheorem{claim}[theorem]{Claim}
-\newtheorem{defn}[theorem]{Definition}
-
-% definition style
-\theoremstyle{definition}
-\newtheorem{problem}{Problem}
-
-% custom
-\newenvironment{solution}{\paragraph{Solution.}}{\qed}
-
-%% custom commands
-\newcommand*{\arXiv}[1]{\href{http://arxiv.org/pdf/#1}{arXiv:#1}}       % arXiv link
-\newcommand*{\eqdef}{\stackrel{\mbox{\normalfont\tiny{def}}}{=}}        % definition by equality
-\newcommand*{\abs}[1]{\lvert #1\rvert}     % absolute values, cardinality
-\newcommand*{\eps}{\varepsilon}            % prettier epsilon
-\newcommand*{\R}{\mathbb{R}}               % real numbers
-\newcommand*{\C}{\mathbb{C}}               % complex numbers
-\newcommand*{\Q}{\mathbb{Q}}               % rational numbers
-\newcommand*{\Z}{\mathbb{Z}}               % integers
-\newcommand*{\N}{\mathbb{N}}               % natural numbers
-\newcommand*{\F}{\mathbb{F}}               % field
-\newcommand*{\family}{\mathcal{F}}         % family
-\newcommand*{\GL}{\mathbf{GL}}             % general linear group
-\newcommand*{\what}[1]{\widehat{#1}}       % wider hat
-\newcommand*{\wbar}[1]{\overline{#1}}      % wider bar
-\newcommand*{\mcal}[1]{\mathcal{#1}}       % mathcal
-\newcommand*{\mbf}[1]{\mathbf{#1}}         % mathbf
-\newcommand*{\mbb}[1]{\mathbb{#1}}         % mathbb
-
-% operators
-\let\Pr\relax\DeclareMathOperator*{\Pr}{\mathbf{Pr}}       % probability
-\DeclareMathOperator*{\E}{\mathbb{E}}          % expectation
-\DeclareMathOperator*{\Var}{\mathbf{Var}}      % variance
-\DeclareMathOperator*{\Span}{span}             % span
-\DeclareMathOperator{\tr}{tr}                  % trace
-\DeclareMathOperator{\rank}{rank}              % rank
-\DeclareMathOperator{\supp}{supp}              % support
-\DeclareMathOperator{\diam}{diam}
-\DeclareMathOperator{\Aut}{Aut}
-
-
-
-\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 \doctype\ \url{\courselink/\assignlink}}
-\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}[2 pts]
-    Let $G$ and $H$ be graphs.
-    A \emph{graph homomorphism} from $G$ to $H$ is a function $f\colon V(G)\to V(H)$ such that $\{f(u),f(v)\}\in E(H)$ whenever $\{u,v\}\in E(G)$.\footnote{Note that a graph homomorphism does \emph{not} need to be a bijection and that it could be the case that $\{f(u),f(v)\}\in E(H)$ even though $\{u,v\}\notin E(G)$.}
-
-    Prove that a graph $G$ is bipartite if and only if there is a graph homomorphism from $G$ to $K_2$.
-\end{problem}
-
-
-\begin{problem}[2 pts]
-    A graph $G$ is called \emph{self-complementary} if $G\cong\wbar G$.
-    For example, $P_4$ and $C_5$ are self-complementary.
-
-    Prove that if $G$ is a self-complementary graph on $n$ vertices, then $n$ is congruent to either $0$ or $1$ modulo $4$.
-\end{problem}
-
-
-\begin{problem}[2 pts]
-    Prove that a graph $G$ on $n$ vertices with $\delta(G)\geq 3$ must contain a cycle of length at most $\lfloor n/2\rfloor+1$.
-\end{problem}
-
-
-\begin{problem}[2 pts]\label{paths}
-    Let $T$ be a tree on $n$ vertices with $\Delta(T)\leq 2$.
-    Prove that $T\cong P_n$.
-\end{problem}
-
-
-
-\begin{problem}[2 pts]
-    Determine (with proof) all trees $T$ (up to isomorphism) on $n\geq 2$ vertices such that $\wbar T$ is also a tree.
-    (Note: we do not require that $T\cong\wbar T$.)
-\end{problem}
-
-
-
-\end{document}