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\section{Graph Theory}

\noindent{\bf{Problem 1:}} Prove that every complete graph with $n>2$ vertices contains a cycle of length $n$.~\\[10pt]

\noindent{\bf{Problem 2:}} Let $G$ be a (not necessarily connected) graph where the length of the longest path is $2$. Prove that $G$ is bipartite.~\\[10pt]

\noindent{\bf{Problem 3:}} How many length $4$ paths exist on a complete graph with $n$ vertices?~\\[10pt]

\noindent{\bf{Problem 4:}} What is the largest number of edges that a graph with $n$ vertices can have and still be 2-colorable?~\\[10pt]

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