Vertex partitions by connected monochromatic k-regular graphs

Generalizing a result of Erdos. Gyarfas and Pyber we show that there exists a constant c such that for any integers k greater than or equal to 2 and for any coloring of the edges of a complete graph with, colors, its vertices can be partitioned into at most r(c(r log r + k)) connected monochromatic k-regular subgraphs and vertices. We also show that the same result holds for complete bipartite graphs, generalizing a result of Haxell. (C) 2000 Academic Press.