The chromatic polynomial of a graph $G$ is the polynomial $C_G(k)$ computed recursively using the theorem of Birkhoff and Lewis. The theorem of Birkhoff and Lewis states: $c_G(k) = c_{G-e}(k) - c_{G/e}(k)$ where $e$ is any edge from $G$, and
- $G - e$ is the graph obtained from $G$ by removing edge $e$.
- $G/e$ the graph obtained from $G$ by removing $e$, identifying the end vertices of $e$, and leaving only one copy of any resulting multiple edges.
Given the graphs $K_{1,3}$ , $K_{1,5}$, $C_4$, $C_5$ and $K_4-e$ , find the chromatic polynomials and determine how many $5$-colorings exist.
Appreciate any help and answers.