Finite Solvable Groups Whose Character Degree Graphs Are Not Complete

In this paper, we characterize the finite solvable groups with non-complete character degree graphs by proving the following theorem, which generalizes a conjecture by Huppert. Suppose that G is a finite solvable group and p is a prime number dividing the degree of some irreducible character of G. If there is another such prime number q such that pq does not divide the degree of any irreducible character of G, then both p-length ℓp(G) and q-length ℓq(G) of G are at most two, and ℓp(G)+ ℓq(G)=4 if and only if pq=6 with QG/Zφ(QG)≅ 32:GL(2,3), where QG is generated by all Sylow 2-subgroups of G and Zφ(G) is a normal nilpotent subgroup of G. Moreover, the bounds are best possible.