A characterization of the prime graphs of solvable groups

Let π(G)π(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G  , denoted ΓGΓG, is the graph with vertex set π(G)π(G) with edges {p,q}∈E(ΓG){p,q}∈E(ΓG) if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle-free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with minimal prime graphs, and we show that solvable groups whose prime graphs are minimal have Fitting length 3 or 4.