Abstract
In 1980, Akiyama and Harary proposed the following problem in the Proceedings-Fourth International Graph Theory Conference: “Are there any graphsG which are not self-complementary withG and\(\bar G\) having the same chromatic polynomial?”. The problem has been unsolved until now. This paper proves that there are no graphsG which are not self-complementary withG and\(\bar G\) having the same chromatic polynomial when |V(G)| =p < 8 orp = 2, 3 (mod 4), there must be a graphG which are not self-complementary withG and\(\bar G\) having the same chromatic polynomial when |V(G)| =p ≥ 8 andp ≡ 0, 1 (mod 4).
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Research supported by the National Natural Science Foundation of China, 19171007
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Xu, J., Liu, Z. The chromatic polynomial between graph & its complement—about Akiyama and Hararys' open problem. Graphs and Combinatorics 11, 337–345 (1995). https://doi.org/10.1007/BF01787814
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DOI: https://doi.org/10.1007/BF01787814