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\begin{document}
\begin{center}
{\bf Math 432 Homework Nine}\\
Due: Saturday, April 2, 2016
\end{center}

\bigskip

Prove the following statements.  Four points for each. \\

\begin{enumerate}


\item Prove that if $G$ is $2$-connected, then $G-xy$ is $2$-connected if and only if $x$ and $y$ lie on a cycle in $G-xy$.  Conclude that a $2$-connected graph is minimally $2$-connected if and only if every cycle is an induced subgraph.  \\

\item Let $G$ be a $k$-connected graph, and let $S,T$ be disjoint subsets of $V(G)$ with size at least $k$. Prove that $G$ has $k$ pairwise disjoint $S,T$-paths.  \\

\item  Suppose that $S$ is a set of $k$ vertices in a $k$-connected graph $G$, with $k\ge 2$.  Prove that $G$ has a cycle containing $S$.  \\

\item Prove that a set of edges in a connected plane graph $G$ forms a spanning tree of $G$ if and only if the duals of the remaining edges form a spanning tree of $G^*$. \\

 
 
\item Use Euler's formula to count the regions formed by $n$ pairwise-crossing lines in the plane, where no three lines have a common point. (Comment: proving the formula by induction is not acceptable. Hint: modify the picture to obtain a finite plane graph.)\\


\item Give two proofs that the Petersen graph is nonplanar.\\
a) Using Kuratowski's Theorem\\
b) Using Euler's Formula and the fact that the Petersen graph has girth $5$. \\


\item Use Euler's Formula and properties of planar graphs, prove that for any planar graph $G$,   $$\sum_{v\in V(G)} (d(v)-4)+\sum_{F\in F(G)} (d(F)-4)=-8,$$ where $F(G)$ is the set of faces of $G$. \\
 

\end{enumerate}
\end{document}
