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\begin{document}
\begin{center}
{\bf Math 432 Homework Ten}\\
Due: Friday, April 8, 2016
\end{center}

\bigskip

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

\begin{enumerate}

\item Show that $\chi(G)=\max\{\chi(B): \text{$B$ a block of $G$}\}$. \\

\item (6 points) A graph is $k$-degenerate if it can be reduced to $K_1$ by repeatedly deleting vertices of degree at most $k$. \\
(a) Show that a graph is $k$-degenerate if and only if every subgraph has a vertex of degree at most $k$.\\
(b) Characterize the $1$-degenerate graphs. \\
(c) Show that every $k$-degenerate graph is $(k+1)$-colorable. \\


\item Let $\overline{G}$ be the complement graph of graph $G$, that is, an edge in $\overline{G}$ if and only if it is not in $G$. Show that $\chi(G)+\chi(\overline{G})\le n(G)+1$. \\

\item Let $G$ be a graph whose odd cycles are pairwise intersecting, meaning that every two odd cycles in $G$ have a common vertex. Prove that $\chi(G)\le 5$.  Construct a graph to show that the bound cannon be improved. \\

\item Prove that if $G$ has no induced $2K_2$, then $\chi(G)\le {\omega(G)+1\choose 2}$. \\

 \item For all $k\in \mathbf{N}$, prove that a graph $G$ is $2^k$-colorable if and only if $G$ can be decomposed to $k$ edge-disjoint bipartite graphs. 
  




\end{enumerate}
\end{document}
