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\begin{document}
\begin{center}
{\bf Math 432 Homework Eleven}\\
Due: Friday, April 15, 2016
\end{center}

\bigskip

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

\begin{enumerate}

\item Prove that every regular graph having a cut-vertex is Class 2 (i.e.,  the chromatic index is $\Delta+1$). \\

\item Let $G$ be a simple graph. \\
(a) Prove that the number of edges in $L(G)$ is $\sum_{v\in V(G)} {d(v)\choose 2}$. \\
(b) Prove that $G$ is isomorphic to $L(G)$ if and only if $G$ is $2$-regular. \\

\item Let $G$ be a bipartite graph with maximum degree $k$. Let $f$ be a proper $k$-edge-coloring of a subgraph $H$ of $G$. Let $uv$ be an edge not in $H$. By using a path alternating in two colors, show that $f$ can be altered and then extended to a proper $k$-edge-coloring of $H+uv$. Conclude that $\chi'(G)=\Delta(G)$. \\


\item Prove that every $2$-connected simple graph $G$ has a cycle of length at least $\min\{n(G), 2\delta(G)\}$. Use this to prove that every $2k$-regular graph with $4k+1$ vertices is Hamiltonian. \\


\item Let $R(p, q)$ be the Ramsey number.  Prove that $R(p,q)\le R(p-1,q)+R(p,q-1)$. \\

\item Find $R(K_3, 2K_2)$. \\



\end{enumerate}
\end{document}
