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

Abstract: Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $\Delta(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists of the prime divisors of the numbers in ${\rm{cd}}(G)$, two distinct vertices $p$ and $q$ being adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. In this note, we provide an upper bound on the size of ${\rm{V}}(G)$ in terms of the clique number $\omega(G)$ (i.e., the maximum size of a subset of ${\rm{V}}(G)$ inducing a complete subgraph) of $\Delta(G)$. Namely, we show that $|{\rm{V}}(G)|\leq{\rm{max}}{2\omega(G)+1,\;3\omega(G)-4}$. 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].