ON THE CHARACTER DEGREE GRAPH OF SOLVABLE GROUPS

Let G be a finite solvable group, and let Delta(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. A fundamental result by P. P. Palfy asserts that the complement (Delta) over bar (G) of the graph Delta(G) does not contain any cycle of length 3. In this paper we generalize Palfy's result, showing that (Delta) over bar (G) does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of Delta(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 Delta(G), then Delta(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 rho-sigma conjecture by B. Huppert.