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. Pálfy asserts that the complement $ ar {Delta }(G)$ of the graph $ Delta (G)$ does not contain any cycle of length $ 3$. In this paper we generalize Pálfy's result, showing that $ ar {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 $ ho $-$ sigma $ conjecture by B. Huppert.