Coloring Semirandom Graphs

We study semirandom k-colorable graphs made up as follows. Partition the vertex set V = {1, . . . , n} randomly into k classes V1, . . . , Vk of equal size and include each Vi-Vj -edge with probability p independently (1 ≤ i < j ≤ k) to obtain a graph G0. Then, an adversary may add further Vi-Vj-edges (i 6= j) to G0, thereby completing the semirandom graph G = G∗n,p,k . We show that if np ≥ max{(1 + ε)k lnn,C0k} for a certain constant C0 > 0 and an arbitrarily small but constant ε > 0, an optimal coloring of G∗n,p,k can be found in polynomial time with high probability. Furthermore, if np ≥ C0 max{k lnn, k}, a k-coloring of G∗n,p,k can be computed in polynomial expected time. Moreover, an optimal coloring of G∗n,p,k can be computed in expected polynomial time if k ≤ ln n and np ≥ C0k lnn. By contrast, it is NP-hard to k-color G∗n,p,k w.h.p. if np ≤ ( 1 2 − ε)k ln(n/k).