Random Graphs Almost Optimally Colorable in Polynomial Time

Let r be an integer ≥3. If the sequence (p n ) satisfies the conditions (1 - p n )n 2/(p) →∞ and (1 - p n )n 2/(p+1) →0, then the random graph G = (n, p n ) on n vertices where each edge is present with probability p n is almost surely colorable in polynomial time using no more than (1+o(1))χ(G) colors. The coloring algorithm used is the greedy algorithm for a maximal system of pairwise disjoint sets applied to a previously computed list of the independent sets of size r in G(n, p n ) .