Colouring Semirandom Graphs

Abstract

We study semirandom k-colourable 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 )klnn, C0k2} for a certain constant C0>0 and an arbitrarily small but constant >0, an optimal colouring of G*n,p,k can be found in polynomial time with high probability. Furthermore, if np ≥ C0max{klnn, k2}, a k-colouring of G*n,p,k can be computed in polynomial expected time. Moreover, an optimal colouring of G*n,p,k can be computed in expected polynomial time if k ≤ ln1/3n and np ≥ C0klnn. By contrast, it is NP-hard to k-colour G*n,p,k With high probability if $np\leq (\frac12-\varepsilon)k\ln(n/k)$ .