Finite groups whose coprime graphs are AT-free

Assume that G is a finite group. The coprime graph of G, denoted by F(G), is an undirected graph whose vertex set is G and two distinct vertices x and y of F(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 F(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.