On the tractability of coloring semirandom graphs
Information Processing Letters
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
| Hi-index | 0.00 |
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 j ≤ k) to obtain a graph G0. Then, an adversary may add further Vi–Vj-edges (i≠j) to G0, thereby completing the semirandom graph G = G*n,p,k. We show that if np ≥ max{(1 + )klnn, C0k2} for a certain constant C00 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 $np\leq (\frac12-\varepsilon)k\ln(n/k)$.