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\centerline{\bf Math 365 -- Wednesday 5/1/19}

\begin{try}\begin{enumerate}[(a)]
\item The chromatic polynomial for the cycle $C_n$ is  $\chi(C_n,t) = (t-1)^n + (-1)^n(t-1)$. 
\begin{enumerate}[(i)]
\item Draw all the ways of coloring the 3-cycle with 3 colors. Then compute $\chi(C_3,3)$ and compare your answers.



\item How many ways are there to color the 5-cycle with 3 colors? 



\item How many ways are there to color the 6-cycle with 2 colors? 


\item Use $\chi(C_n,t)$ to verify that even cycles are bipartite and odd cycles are not. 
\end{enumerate}

\item For $G$ and $H$ below, compute the number of ways to color the graph with a palate of 1 color, of 2 colors, of 3 colors, and of 4 colors. For $K$, compute the number of ways to color the graph with a palate of 1 color, of 2 colors, of 3 colors, of 4 colors, and of 5 colors.  
$$
G = \begin{matrix}\begin{tikzpicture}[scale = 1.5]
\node[bV, label=above:{$a$}] (a) at (0,1){};
\node[bV, label=above:{$b$}] (b) at (1,1){};
\node[bV, label=below:{$c$}] (c) at (0,0){};
\node[bV, label=below:{$d$}] (d) at (1,0){};
\draw (a) to (b) to (c) to (a) (b) to (d);
\end{tikzpicture}\end{matrix}\qquad \qquad 
H = \begin{matrix}\begin{tikzpicture}
\node[bV, label=above:{$a$}] (a) at (1,1){};
\node[bV, label=left:{$b$}] (b) at (0,0){};
\node[bV, label=right:{$c$}] (c) at (2,0){};
\node[bV, label=below:{$d$}] (d) at (1,-1){};
\draw (a) to (b) to (d) to (c) to (a) to (d);
\end{tikzpicture}\end{matrix}\qquad \qquad 
K = \begin{matrix}\begin{tikzpicture}
\node[bV, label=above:{$a$}] (a) at (1,1){};
\node[bV, label=below:{$b$}] (b) at (1,-1){};
\node[bV, label=below:{$c$}] (c) at (2,0){};
\node[bV, label=below:{$d$}] (d) at (3,-1){};
\node[bV, label=above:{$e$}] (e) at (3,1){};
\draw (c) to (a) to (b) to (c) to (d) to (e) to (c);
\end{tikzpicture}\end{matrix}
$$
{[Hint: For all three, break into cases based on how many colors you actually use. Then for any coloring using exactly $k$ colors, there will be $k!$ colorings with that ``color pattern'' -- so instead of listing all colorings, list color patterns and then account for how many colorings correspond to each pattern. For $K$, you might want to use some of what you've already computed  about colorings of $C_3$ to count colorings of $K$ efficiently.]}




\item Calculate the chromatic polynomial for $G$ and for $H$. 

\item Explain why  $\chi(K_n,t) = t(t-1)(t-2) \cdots (t - (n-1))$.
\end{enumerate}
\end{try}



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