On The Enhanced Power Graph of a Semigroup

TL;DR

This paper explores the structure and properties of the enhanced power graph of a semigroup, providing characterizations of its connectivity, special graph types, and chromatic number, including a counterexample for uncountable chromatic number.

Contribution

It offers a comprehensive structural analysis of the enhanced power graph of semigroups, including characterizations of various graph properties and a counterexample for the chromatic number.

Findings

Characterized when the enhanced power graph is complete, bipartite, regular, a tree, or null.

Analyzed the connectedness, planarity, and degrees of the graph.

Constructed an example with uncountably infinite chromatic number.

Abstract

The enhanced power graph $\mathcal P_e(S)$ of a semigroup $S$ is a simple graph whose vertex set is $S$ and two vertices $x,y \in S$ are adjacent if and only if $x, y \in \langle z \rangle$ for some $z \in S$, where $\langle z \rangle$ is the subsemigroup generated by $z$. In this paper, first we described the structure of $\mathcal P_e(S)$ for an arbitrary semigroup $S$. Consequently, we discussed the connectedness of $\mathcal P_e(S)$. Further, we characterized the semigroup $S$ such that $\mathcal P_e(S)$ is complete, bipartite, regular, tree and null graph, respectively. Also, we have investigated the planarity together with the minimum degree and independence number of $\mathcal P_e(S)$. The chromatic number of a spanning subgraph, viz. the cyclic graph, of $\mathcal P_e(S)$ is proved to be countable. At the final part of this paper, we construct an example of a semigroup $S$ such that the chromatic number of $\mathcal P_e(S)$ need not be countable.