On the character degree graph of solvable groups

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.