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%% header definitions
\def\course{MATH 314}
\def\assign{HW \#4}
\def\due{Feb 8}
\def\doclink{\url{https://mathematicaster.org/teaching/graphs2022/hw/hw4.tex}}

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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}
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\begin{minipage}{.33\textwidth}
	\flushright {\bf \due}
\end{minipage}
\vskip3pt
\hrule\hrule\hrule
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\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}[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}
