On the character degree graph of finite groups

Given a finite groupG, let cd(G)denote the set of degrees of the irreducible complexcharacters ofG.Thecharacter degree graphofGis defined as the simple undirected graphwhose vertices are the prime divisors of the numbers in cd(G), two distinct verticespandqbeing adjacent if and only ifpqdivides some number in cd(G). In this paper, we consider thecomplement of the character degree graph, and we characterize the finite groups for whichthis complement graph is not bipartite. This extends the analysis of Akhlaghi et al. (Proc AmMath Soc 146:1505–1513,2018), where the solvable case was treated