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 $C_n$.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2019
- DOI:
- 10.48550/arXiv.1907.04320
- arXiv:
- arXiv:1907.04320
- Bibcode:
- 2019arXiv190704320L
- Keywords:
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- Mathematics - General Mathematics;
- 05C15;
- 05C30
- E-Print:
- 7 pages, 5 figures