  Let \pi(G) denote the set of prime divisors of the order of a finite group G.
The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} 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.
