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\begin{document}
\begin{center}
{\bf Math 432 Homework Eight}\\
Due: Friday, March 25, 2010
\end{center}

\bigskip

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

\begin{enumerate}
\item The defect $df (S)$ of a set $S\subseteq X$ in an $X, Y$-bigraph is defined to be $|S|-|N(S)|$. The matching number  $\alpha'(G)$ is defined to be the maximum size of a matching in $G$.  Prove that in an $X, Y$-bigraph $G$,  $\alpha'(G)= |X|-\max_{S\subseteq X}\{df (S)\}$. \\

\item Use the Konig-Egervary Theorem to prove that every subgraph of $K_{n,n}$ with more than $(k-1)n$ edges has a matching of size at least $k$. \\

\item Suppose that $G$ is an $r$-connected graph of even order having no $K_{1, r+1}$ as an induced subgraph. Prove that $G$ has a perfect matching. \\


\item Prove that $\kappa'(G)=\kappa(G)$ if $G$ is a $3$-regular simple graph. Find (with proof) the smallest $3$-regular simple graph having connectivity $1$. \\



\item Let $H$ be the block-cutpoint graph of a graph $G$ that has a cut-vertex. \\
(a) prove that $H$ is a forest. \\
(b) Prove that $G$ has at least two blocks that contain one cut-vertex of $G$. \\
(c) Prove that every graph has fewer cut-vertices than blocks. \\

\item Suppose that $G$ has no isolated vertices.  Prove that if $G$ has no even cycles, then every block of $G$ is an edge or an odd cycle. \\

  

\end{enumerate}
\end{document}
