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%% header definitions
\def\course{MATH 314}
\def\assign{HW \#2}
\def\due{Feb 1}
\def\doclink{\url{https://mathematicaster.org/teaching/graphs2022/hw/hw2.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}
\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}[1 pt]
    Recall that a \emph{directed graph} (or digraph) is a pair $D=(V,E)$ where $V$ is a set and $E\subseteq V^2$.
    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)
\end{problem}





\end{document}
