Classifying prime graphs of finite groups -- a methodical approach

For a finite group G, the vertices of the prime graph (G) are the primes that divide |G|, and two vertices p and q are connected by an edge if and only if there is an element of order pq in G. Prime graphs of solvable groups as well as groups whose noncyclic composition factors have order divisible by exactly three distinct primes have been classified in graph-theoretic terms. In this paper, we begin to develop a general theory on the existence of edges in the prime graph of an arbitrary T-solvable group, that is, a group whose composition factors are cyclic or isomorphic to a fixed nonabelian simple group T. We then apply these results to classify the prime graphs of T-solvable groups for, in a suitable sense, most T such that |T| has exactly four prime divisors. We find that these groups almost always have a 3-colorable prime graph complement containing few possible triangles.