a) Let $n \in \mathbb{Z}^+$ represent the number of colors necessary to appropriately color the vertices of the graph $K_{2,3}$ in the figure below.
(i) In how many instances of $K_{2,3}$'s proper colorings are vertices a and b the same color?
(ii) How many proper colorings of $K_{2,3}$ have vertices a, b colored with different colors?
b) Provide the chromatic polynomial for $K_{2,3}$. What is $\chi(K_{2,3})$?
c) For $n \in \mathbb{Z}^+$, provide the chromatic polynomial for $K_{2,n}$. What is $\chi(K_{2,n})$?