Classifying prime graphs of finite groups -- a methodical approach

For a finite group G, the vertices of the prime graph Γ(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.