Monochromatic bounded degree subgraph partitions

Let F={F1,F2,…}F={F1,F2,…} be a sequence of graphs such that FnFn is a graph on nn vertices with maximum degree at most ΔΔ. We show that there exists an absolute constant CC such that the vertices of any 2-edge-colored complete graph can be partitioned into at most 2CΔlogΔ2CΔlogΔ vertex disjoint monochromatic copies of graphs from FF. If each FnFn is bipartite, then we can improve this bound to 2CΔ2CΔ; this result is optimal up to the constant CC.