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\headers{Math378}{Homework 13}{Due Fri. Apr 25, 2025}

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\noindent
\textbf{Directions:} Solve the following problems.  All written work must be your own.  See the course syllabus for detailed rules.

\begin{enumerate}

\item {} [10.1.13] Prove that $d_1,\ldots,d_p$ is graphic if and only if $p-1-d_1,\ldots,p-1-d_p$ is graphic.

\item {} [10.2.1] Let $x$ and $y$ be vertices of a graph.  
\begin{enumerate}
	\item Suppose that there is a closed walk containing both $x$ and $y$.  Must there be a closed trail containing both $x$ and $y$?
	\item Suppose that there is a closed trail containing both $x$ and $y$.  Must there be a cycle containing both $x$ and $y$?
\end{enumerate}

\item {} [10.2.8] Let $G$ be a forest consisting of $t$ trees, and let $n=|V(G)|$.  How many edges does $G$ have?  Prove your answer is correct.

\item {} [10.2.17] Let $n\ge 2$ and let $d_1\ge d_2 \ge \cdots \ge d_n$ be a sequence of positive integers.  Prove that this sequence is the degree sequence of a tree if and only if $\sum_{i=1}^n d_i = 2n-2$.

\end{enumerate}
\end{document}
