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,(G) and q-length l(q)(G) of G are at most two, and l(p)(G) + l(q)(G) = 4 if and only if pq = 6 with Q(G)/Z phi(Q(G)) congruent to 3(2) : GL(2,3), where Q(G) is generated by all Sylow 2-subgroups of G and Z(phi)(G) is a normal nilpotent subgroup of G. Moreover, the bounds are best possible.Mathematics, AppliedMathematicsSCI(E)中国科学引文数据库(CSCD)6ARTICLE4541-5521