% Chris Hanusa
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\begin{center}\large 
MATH 634, Spring 2014

{\sc Homework 8}

to be prepared for presentation at  5:00{\sc pm} on Monday, March 3.
\end{center}

\noindent
{\em Background reading:} {\em Pearls in Graph Theory}, Sections 2.1 through 3.1.


\begin{enumerate}
\item[\bf 8-1.]
\begin{enumerate}
\item Suppose that $P$ and $Q$ are two {\bf maximum} paths in a connected graph $G$. (That is, no other path in $G$ has longer length.) Prove that $P$ and $Q$ {\bf must} share a common vertex.
\item Suppose that $P$ and $Q$ are two {\bf maximal} paths in a connected graph $G$. (That is, neither is contained in any longer path.)  Show that $P$ and $Q$ {\bf might not} share a common vertex. 
\end{enumerate}

\item[\bf 8-2.]
\begin{enumerate}
\item Find a graph $E$ which has an Eulerian circuit but no Hamiltonian cycle.
\item Find a graph $H$ which has a Hamiltonian cycle but no Eulerian circuit.
\end{enumerate}
[{\em If either is impossible, prove why you can not find such a graph.}]

\item[\bf 8-3.]
Which of the graphs in Figure 1.2.5 are Hamiltonian?

\item[\bf 8-4.]
\begin{enumerate}
\item Prove that there is no closed knight's tour on the $3\times8$ grid, without simply citing the theorem from class.
\item Find a closed knight's tour on the $3\times 5$ {\bf torus}.
\end{enumerate}

\item[\bf 8-5.]
Is is possible to decompose $W_n$ into two spanning trees?  Prove your answer.


\end{enumerate}

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