On the character degree graph of solvable groups

Abstract:

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 $\bar {\Delta }(G)$ of the graph $\Delta (G)$ does not contain any cycle of length $3$. In this paper we generalize Pálfy’s result, showing that $\bar {\Delta }(G)$ 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 $\rho$-$\sigma$ conjecture by B. Huppert.