Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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THE CHROMATIC POLYNOMIAL FOR CYCLE GRAPHS

  • Received : 2019.05.17
  • Accepted : 2019.06.17
  • Published : 2019.06.30
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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 $n{\geq}1$. Also its inductive proof is widely well-known by the 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

E1KKMK_2019_v27n2_525_f0001.png 이미지

FIGURE 2. Cn (1 ≤ n ≤ 5)

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FIGURE 3. Cn+1 , Pn+1 and Cn

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FIGURE 4. A cycle graph C5 and a graph K4 with names of colors

E1KKMK_2019_v27n2_525_f0004.png 이미지

FIGURE 5. A graph G and its adjacency matrix A

E1KKMK_2019_v27n2_525_f0005.png 이미지

FIGURE 1. G , G - e and G/e

References

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  2. Ronald C. Read, An introduction to chromatic polynomials, J. Combinatorial Theory 4 (1968), 52-71. MR0224505. https://doi.org/10.1016/S0021-9800(68)80087-0
  3. Hassler Whitney, Congruent Graphs and the Connectivity of Graphs, Amer. J. Math. 54 (1) (1932), 150-168. MR1506881. https://doi.org/10.2307/2371086