The chromatic polynomial for cycle graphs

Abstract

Let $P(G,\lambda )$ denote the number of proper vertex colorings of $G$ with $\lambda$ colors. The chromatic polynomial $P(C_n,\lambda )$ for the cycle graph $C_n$ is well-known as
$$P(C_n,\lambda )=(\lambda -1{)}^n+(-1{)}^n(\lambda -1)$$
for all positive integers $n\ge 1$. Also its inductive proof is widely well-known by the \emph{deletion-contraction recurrence}. In this paper, we give this inductive proof again and three other proofs of this formula of the chromatic polynomial for the cycle graph $C_n$.