Existence of solvable groups with prescribed two-component character degree graphs

Determine whether, for every pair of integers n,N>1 satisfying N\geq 2n-1, there exists a finite solvable group G whose character degree graph \Delta(G) has exactly two connected components with cardinalities n and N, respectively.

Background

For a finite solvable group G, the character degree graph \Delta(G) has vertices given by the primes dividing irreducible character degrees, with two primes joined when their product divides some character degree. General structure theorems show that \Delta(G) has at most two connected components, and the sizes of those components are subject to the constraint N\geq 2n-1 when the components have cardinalities n and N.

The paper records the unresolved existence question for all admissible pairs (n,N). It notes that no pair is currently known for which the desired solvable group fails to exist, and gives a construction realizing every case in which one component is an isolated vertex and the other has arbitrary prescribed size N. The broader question of realizing every pair of component sizes remains unresolved.