Korean J. Math. Vol. 27 No. 2 (2019) pp.525-534
DOI: https://doi.org/10.11568/kjm.2019.27.2.525

The chromatic polynomial for cycle graphs

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Jonghyeon Lee
Heesung Shin


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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Supporting Agencies

National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2017R1C1B2008269).


[1] George D. Birkho , A determinant formula for the number of ways of coloring a map, Ann. of Math. (2) 14 (1912/13), no. 1-4, 42–46. MR1502436 Google Scholar

[2] Ronald C. Read, An introduction to chromatic polynomials, J. Combinatorial Theory 4 (1968), 52–71. MR0224505 Google Scholar

[3] Hassler Whitney, Congruent Graphs and the Connectivity of Graphs, Amer. J. Math. 54 (1932), no. 1, 150–168. MR1506881 Google Scholar