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\def\assignment{HW \#4}
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\def\due{Due on Feb 14}




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\begin{document}
\noindent  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.
\vskip10pt


\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 $\{u,v\}\in E(G)$. 
Prove that a graph $G$ is bipartite if and only if there is a graph homomorphism from $G$ to $K_2$. 

(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 $\{u,v\}\notin E(G)$)
\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]\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}


\begin{problem}[2 pts]Let $G$ and $H$ be two self-complementary graphs on disjoint vertex sets, where $H$ has even order $n$. 
	Let $F$ be the graph obtained from $G\cup H$ by joining each vertex of $G$ to every vertex of degree less than $n/2$ in $H$. 
	Show that $F$ is self-complementary.  
	
	\emph{Hint}: A graph $F$ is self-complementary if there exists a $\varphi : V(F) \rightarrow V(F)$ such that $vw\in E(F)$ if and only if $\varphi(v)\varphi(w)\not\in E(F)$.
\end{problem}



\end{document}

Let $G$ be a connected graph and let $S$ be any subset of edges of $G$. Show that  $G$ has a connected, spanning subgraph $H$ such that $S\subseteq E(H)$ and if $C$ is a cycle in $H$, then $E(C) \subseteq S$.