Clique partitions of complements of forests and bounded degree graphs

In this paper, we prove that for any forest F⊂KnF⊂Kn, the edges of E(Kn)⧹E(F)E(Kn)⧹E(F) can be partitioned into O(nlogn)O(nlogn) cliques. This extends earlier results on clique partitions of the complement of a perfect matching and of a hamiltonian path in KnKn.In the second part of the paper, we show that for n   sufficiently large and any ε∈(0,1]ε∈(0,1], if a graph G   has maximum degree O(n1-ε)O(n1-ε), then the edges of E(Kn)⧹E(G)E(Kn)⧹E(G) can be partitioned into O(n2-(1/2)εlog2n) cliques provided there exist certain Steiner systems. Furthermore, we show that there are such graphs G   for which Ω(ε2n2-2ε)Ω(ε2n2-2ε) cliques are required in every clique partition of E(Kn)⧹E(G)E(Kn)⧹E(G).