Finite groups whose coprime graphs are AT-free

Abstract

Assume that $ G $ is a finite group. The coprime graph of $ G $, denoted by $ \Gamma(G) $, is an undirected graph whose vertex set is $ G $ and two distinct vertices $ x $ and $ y $ of $ \Gamma(G) $ are adjacent if and only if $ (o(x), o(y)) = 1 $, where $ o(x) $ and $ o(y) $ are the orders of $ x $ and $ y $, respectively. This paper gives a characterization of all finite groups with AT-free coprime graphs. This answers a question raised by Swathi and Sunitha in Forbidden subgraphs of co-prime graphs of finite groups. As applications, this paper also classifies all finite groups $ G $ such that $ \Gamma(G) $ is AT-free if $ G $ is a nilpotent group, a symmetric group, an alternating group, a direct product of two non-trivial groups, or a sporadic simple group.

Keywords

• coprime graph
• at-free graph
• finite group