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\begin{document}
\begin{center}
{\bf Math 432 Homework Seven}\\
Due: Saturday, March 19, 2016
\end{center}

\bigskip

Prove the following statements.  Four points for each. 

\begin{enumerate}
\item Prove that Petersen graph has no cycle of length $7$ or $10$.\\

\item Let $G$ be the $3$-regular graph with $4m$ vertices formed from $m$ pairwise disjoint kites ($K_4$ minus an edge) by adding $m$ edges to link them in a ring. Prove that it has $2m\cdot 8^m$ spanning trees. \\

\item (6 points) Count the following sets of trees with vertex set $[n]=\{1,2,\ldots, n\}$, given two proofs for each: one using the Prufer correspondence and one by direct counting arguments.\\
(a) trees that have $2$ leaves.\\
(b) trees that have $n-2$ leaves. \\



\item Prove that a $d$-regular simple graph $G$ has a decomposition into copies of $K_{1,d}$ if and only if it is bipartite. \\



\item Prove that a graph is bipartite if and only if every subgraph $H$ of $G$ has an independent set consisting of at least half of $V(H)$, where a set of vertices is independent if there is no edge between the vertices in the set. \\


\item Let $A=(A_1, \ldots, A_m)$ be a collection of subsets of a set $Y$. A system of distinct representatives (SDR) for $A$ is a set of distinct elements $a_1, a_2, \ldots, a_m$ in $Y$ such that $a_i\in A_i$.  Prove that $A$ has an SDR if and only if $|\cup_{i\in S} A_i|\ge |S|$ for every $S\subseteq \{1, 2, \ldots, m\}$.



\end{enumerate}
\end{document}
