Department of Mathematics

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Chromatic Number of the Cyclic Graph of Infinite Semigroup
(ACM Digital Library, 2020-01) Kumar, Jitender
The cyclic graph Γ(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 Γ(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).
On the clique number and independence number of the cyclic graph of a semigroup
(World Scientific, 2023-04) Kumar, Jitender
The cyclic graph Γ(S) of a semigroup S is the simple undirected graph whose vertex set is S and two vertices x,y are adjacent if the subsemigroup generated by x and y is monogenic. In this paper, we determine the clique number of Γ(S) for an arbitrary semigroup S. Further, we obtain the independence number of Γ(S) if S is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of Γ(S) if S is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds.