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\begin{document}
\title{Math 101 Homework}
\author{Mary Radcliffe}
\date{due 20 Jan 2015}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item A graph $G=(V, E)$ is called {\it bipartite} if the vertex set can be written as $V=A\cup B$, where $A$ and $B$ are disjoint, and every edge has one endpoint in $A$ and the other in $B$.

Let $G$ be a graph. Prove that $G$ is 2-chromatic if and only if $G$ is bipartite.

\item Determine the chromatic number of the Petersen graph.

\item A {\it cycle} on $n$ vertices (also called an $n$-cycle, denoted $C_n$) is the graph you get by considering an $n$-gon, with the vertices of the $n$-gon the vertices of the graph, and the edges of the $n$-gon the edges of the graph. For example, a $6$-cycle and a $7$-cycle are shown below.


Determine $\chi(C_n)$ for any $n$. 

\begin{figure}[htp]\centering
\includegraphics[width=.8\textwidth]{pic.pdf}
\caption{$C_6$ (left) and $C_7$ (right)}
\end{figure}


\end{enumerate}

\end{document}