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\def\assignment{HW \#2}
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\def\due{Due on Jan 31}




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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}[1 pt]
	Recall that a \emph{directed graph} (or digraph) is a pair $D=(V,E)$ where $V$ is a set and $E\subseteq \{(u,v) \in V\times V \colon u\ne v\}$.
	For $u,v\in V$, a $u$-$v$ diwalk in $D$ is a sequence $(u=v_0,\dots,v_k=v)$ such that $(v_i,v_{i+1})\in E$ for all $i\in\{0,\dots,k-1\}$.
	
	Consider the relation $R$ on $V$ where $u\mathbin R v$ iff there is a $u$-$v$ diwalk.
	Give an example of a digraph $D$ where $R$ is \textbf{not} an equivalence relation.
	Justify your answer.
	(Feel free to draw a picture of $D$)
\end{problem}


\begin{problem}[2 pts]
	Let $G$ be a connected graph and consider a function $f\colon V(G)\to X$ where $X$ is some arbitrary set.
	Prove that if $f$ is \emph{not} a constant function, then there is an edge $uv\in E(G)$ such that $f(u)\neq f(v)$.
\end{problem}



\begin{problem}[2 pts]
	Prove that every graph on at least two vertices has a pair of vertices with the same degree.
\end{problem}


\begin{problem}[2 pts]
	Prove that if $G$ is a graph with $\delta(G)\geq 2$, then $G$ must contain a cycle.
\end{problem}


\begin{problem}[3 pts]
	Let $G$ be a graph and let $A$ be an independent set of $G$.
	Prove that
	\[
	\sum_{v\in A}\deg v \leq\abs{E(G)}
	\]
	with equality if and only if $G$ is bipartite with parts $A$ and $V(G)\setminus A$.
	
	\noindent (Especially in this problem, be sure to carefully justify all steps in your argument)
	
	\noindent \textbf{Hint:} consider the sets $E_v=\{\text{edges incident with } v\}$ and $\widehat{E}=\bigcup_{v \in A} E_v$. For the equality, decide when is $\widehat{E}=E(G)$.
\end{problem}

\end{document}