On the Enhanced Power Graph of a Semigroup

Abstract

The enhanced power graph [Formula: see text] of a semigroup [Formula: see text] is a simple graph whose vertex set is [Formula: see text] and two vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text], [Formula: see text] for some [Formula: see text], where [Formula: see text] is the subsemigroup generated by [Formula: see text]. In this paper, we first describe the structure of [Formula: see text] for an arbitrary semigroup [Formula: see text], and then discuss the connectedness of [Formula: see text]. Further, we characterize the semigroup [Formula: see text] in the cases when [Formula: see text] is separately a complete, bipartite, regular, tree and null graph. The planarity, together with the minimum degree and independence number, of [Formula: see text] is also investigated. The chromatic number of a spanning subgraph, i.e., the cyclic graph, of [Formula: see text] is proved to be countable. In the final part of this paper, we construct an example of a semigroup [Formula: see text] such that the chromatic number of [Formula: see text] need not be countable.