On the character degree graph of solvable groups

Let G be a finite solvable group, and let ∆(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. A fundamental result by P. P. Pálfy asserts that the complement ∆(G) of the graph ∆(G) does not contain any cycle of length 3. In this paper we generalize Pálfy’s result, showing that ∆(G) does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of ∆(G) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if n is the clique number of ∆(G), then ∆(G) has at most 2n vertices. This confirms a conjecture by Z. Akhlaghi and H. P. Tong-Viet, and provides some evidence for the famous ρ-σ conjecture by B. Huppert.