% Chris Hanusa
% chanusa@qc.cuny.edu	
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\begin{center}\large 
MATH 634, Spring 2014

{\sc Homework 13}

to be prepared for presentation at  5:00{\sc pm} on Monday, April 7.
\end{center}

\noindent
{\em Background reading:} {\em Pearls in Graph Theory}, Sections 8.1--8.3, 9.1, 9.2, and 10.1.


\begin{enumerate}
\item[\bf 13-1.]
Find and prove an ``Euler's formula" for {\bf disconnected} planar graphs.

\item[\bf 13-2.]
 This question has to do with planar duality: 
\begin{enumerate}
\item Show that for all $n$, the wheel graph $W_n$ is self-dual. 
\item Find a graph that has two non-isomorphic planar duals. \newline [{\em Hint: Look for different planar embeddings.}]
\end{enumerate}


\item[\bf 13-3.]
Prove that the graph $G$ in Figure 9.1.18 (p.\ 189) is non-planar using two methods: 
\begin{enumerate}
\item Find a subdivision of $K_{3,3}$ or $K_5$ that is a subgraph of $G$. 
\item Through a series of edge deletions and edge contractions, show that either $K_{3,3}$ or $K_5$ is a minor of $G$. 
\end{enumerate}

\item[\bf 13-4.]
Show that the Petersen graph is a minor of this graph:

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\includegraphics[height=1.5in]{figures/Tietzes}
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\item[\bf 13-5.] 
9.1.1ab

\item[\bf 13-6.]
\begin{enumerate}
\item Show that $\theta(K_7)=2$.
\item Show that genus$(K_6)=2$
\end{enumerate}




\end{enumerate}


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