Vertex Partitions by Connected Monochromatic k-Regular Graphs

Generalizing a result of Erdős, Gyárfás and Pyber we show that there exists a constant c such that for any integers r, k⩾2 and for any coloring of the edges of a complete graph with r colors, its vertices can be partitioned into at most rc(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.