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arXiv · 1907.04320

The chromatic polynomial for cycle graphs

Jonghyeon Lee·Heesung Shin
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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 $C_n$.

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Jonghyeon Lee, Heesung Shin. 2019-07-09. The chromatic polynomial for cycle graphs. https://doi.org/10.11568/kjm.2019.27.2.525

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