DOI: https://doi.org/10.11568/kjm.2019.27.2.525
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 $C_n$.
Keywords
Subject classification
05C15, 05C30Sponsor(s)
National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2017R1C1B2008269).Full Text:
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