On the character degree graph of solvable groups

Abstract: Let (G) be a finite solvable group, and let (\Delta(G)) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of (G). A fundamental result by P.P. P\'alfy asserts that the complement $\bar{\Delta}(G)$ of the graph (\Delta(G)) does not contain any cycle of length (3). In this paper we generalize P\'alfy's result, showing that $\bar{\Delta}(G)$ 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 \emph{(\rho)-(\sigma) conjecture} by B. Huppert.