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 wellknown as $$P(C_n,\lambda) = (\lambda1)^n+(1)^n(\lambda1)$$ for all positive integers $n\ge 1$. Also its inductive proof is widely wellknown by the \emph{deletioncontraction 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 $C_n$.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 DOI:
 10.48550/arXiv.1907.04320
 arXiv:
 arXiv:1907.04320
 Bibcode:
 2019arXiv190704320L
 Keywords:

 Mathematics  General Mathematics;
 05C15;
 05C30
 EPrint:
 7 pages, 5 figures