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\begin{center}\textbf{Problem Set 1 \hspace{.7in} Graph Theory \hspace{.7in} Due 9/17/15} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}
\item Do Problem A24 from Marcus.

\item Draw the complement of the graph from Problem A24 in Marcus.

\item Draw a graph with degree sequence (6,5,3,3,2,1,1,1) or explain why no such graph exists.

\item Do Problem A48 from Marcus.

\item Do Problem B10 from Marcus.

\item Do Problem B15 from Marcus.

\item Suppose \textbf{G} is a bipartite graph.  To what extent are the sets \textbf{X} and \textbf{Y} uniquely determined?

\item Find a 4-regular bipartite graph with 16 vertices, or show why one cannot exist.

\item Do Problem D30 from Marcus.

\item If a graph is 4-regular, what can you say about the number of vertices, and why?

\end{enumerate}

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