Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs

We prove that for all integers Delta, r >= 2, there is a constant C = C(Delta, r) > 0 such that the following is true for every sequence F = {F-1, F-2, ...} of graphs with v(F-n) = n and Delta(F-n) <= Delta, for each n is an element of N. In every r-edge-coloured K-n, there is a collection of at most C monochromatic copies from F whose vertex-sets partition V (K-n). This makes progress on a conjecture of Grinshpun and S & aacute;rk & ouml;zy.