Clique partitions of complements of forests and bounded degree graphs

In this paper, we prove that for any forest F subset of K-n, the edges of E(K-n)\E(F) can be partitioned into O(n log n) cliques. This extends earlier results on clique partitions of the complement of a perfect matching and of a hamiltonian path in K-n. In the second part of the paper, we show that for n sufficiently large and any epsilon epsilon (0, 1], if a graph G has maximum degree O(n(1-epsilon)), then the edges of E(K-n)\E(G) can be partitioned into O(n(2-(1/2)epsilon)log(2) n) cliques provided there exist certain Steiner systems. Furthermore, we show that there are such graphs G for which Omega(epsilon(2)n(2-2 epsilon)) cliques are required in every clique partition of E(K-n)\E(G). (C) 2007 Elsevier B.V. All rights reserved.