Bounding the number of vertices in the degree graph of a finite group

Let G be a finite group, and let cd(G) denote the set of degrees of the irreducible complex characters of G. The degree graph Δ(G) of G is defined as the simple undirected graph whose vertex set V(G) consists of the prime divisors of the numbers in cd(G), two distinct vertices p and q being adjacent if and only if pq divides some number in cd(G). In this note, we provide an upper bound on the size of V(G) in terms of the clique number ω(G) (i.e., the maximum size of a subset of V(G) inducing a complete subgraph) of Δ(G). Namely, we show that |V(G)|≤max{2ω(G)+1,3ω(G)−4}. Examples are given in order to show that the bound is best possible. This completes the analysis carried out in [1] where the solvable case was treated, extends the results in [3,4,9], and answers a question posed by the first author and H.P. Tong-Viet in [4