The prime graph on class sizes of a finite group has a bipartite complement

Let G be a finite group, and let cs(G) denote the set of sizes of the conjugacy classes of G. The prime graph built on cs(G), that we denote by Δ(G), is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cs(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in cs(G). A rephrasing of the main theorem in [8] is that the complement Δ‾(G) of the graph Δ(G) does not contain any cycle of length 3. In this paper we generalize this result, showing that Δ‾(G) does not contain any cycle of odd length, i.e., it is a bipartite graph. In other words, the vertex set V(G) of Δ(G) is covered by two subsets, each inducing a complete subgraph (a clique). As an immediate consequence, setting ω(G) to be the maximum size of a clique in Δ(G), the inequality |V(G)|≤2ω(G) holds for every finite group G.