Monochromatic -Decompositions of Graphs

Given graphs G and H, and a colouring of the edges of G with k colours, a monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic graph isomorphic to H. Let ϕ k ( n , H ) be the smallest number ϕ such that any k-edge-coloured graph G of order n, admits a monochromatic H-decomposition with at most ϕ parts. Here we study the function ϕ k ( n , K r ) for k ⩾ 2 and r ⩾ 3 .