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\def\assignment{HW \#1}
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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 $S$ be a finite set of integers and let $G$ be the graph on vertex set $S$ where for any $x,y\in S$, $xy\in E(G)$ if and only if $x+y$ is odd.
\begin{enumerate}
\item Prove that $G$ is a bipartite graph for any such $S$.
\item If $S=\{1, 2, \dots, 100\}$, what is $\abs{E(G)}$? Justify your answer.
\end{enumerate}
\end{problem}


\begin{problem}[2 pts]
Suppose that $(u=v_0,v_1,\dots,v_k=v)$ is a $u$--$v$ geodesic.
Prove that $d(u,v_i)=i$ for all $i\in\{0,\dots,k\}$.
\end{problem}


\begin{problem}[2 pts]\label{abpath}
Let $G$ be a graph.
For two non-empty subsets $A,B\subseteq V(G)$, an $A$--$B$ path is a path in $G$ which connects some vertex of $A$ to some vertex of $B$.
Prove that if $P$ is a minim\textbf{al} $A$--$B$ path, then $P$ contains exactly one vertex from $A$ and contains exactly one vertex from $B$.
\end{problem}


\begin{problem}[2 pts]
Let $G$ be a connected graph.
Prove that any two maxim\textbf{um} paths in $G$ must share some vertex.
\end{problem}


\begin{problem}[2 pts]
Prove that a graph $G$ is bipartite if and only if every subgraph $H$ of $G$ has an independent set consisting of at least half of $V(H)$.
(recall that an independent set is a set of vertices which induce no edges)
\end{problem}


\end{document}