Chromatic Number of the Cyclic Graph of Infinite Semigroup
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The cyclic graph \(\Gamma (S)\) of a semigroup S is the simple graph whose vertex set is S, two element being adjacent if the subsemigroup generated by these two elements is monogenic. The purpose of this note is to prove that the chromatic number of \(\Gamma (S)\) is at most countable. The present paper generalizes the results of Shitov (Graphs Comb 33(2):485–487, 2017) and the corresponding results on power graph and enhanced power graph of groups obtained by Aalipour et al. (Electron J Comb 24(3):#P3.16, 2017).
KeywordsMonogenic semigroup Cyclic graph Power graph Enhanced power graph
Mathematics Subject Classification05C25
The authors are very grateful to the referees for their valuable suggestions which lead to an improvement of this paper. The second author wishes to acknowledge the support of MATRICS Grant (MTR/2018/000779) funded by SERB.