On the character degree graph of solvable groups. I: Three primes (Q1307421)

The character degree graph \(\Gamma(G)\) of a finite group is defined as follows: The vertices are those primes \(p\) that divide \(\chi(1)\) for some irreducible character \(\chi\) of \(G\). Two different vertices \(p,q\in\Gamma(G)\) are connected by an edge if there exists an irreducible character \(\chi\) with \(pq\mid\chi(1)\). Suppose that \(G\) is solvable. \textit{O. Manz, W. Willems} and \textit{T. R. Wolf} [J. Reine Angew. Math. 402, 181-198 (1989; Zbl 0678.20002)] have proved that whenever \(\pi\) is a subset of the vertices of \(\Gamma(G)\) with \(|\pi|=4\) then there exists a \(\chi\in\text{Irr}(G)\) which is divisible by at least 2 primes in \(\pi\). In the paper under review this result has been improved by showing the same conclusion but assuming only \(|\pi|=3\).