Monochromatic Clique Decompositions of Graphs

Let $G$ be a graph whose edges are coloured with $k$ colours, and $mathcal H=(H_1,dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $mathcal H$-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 copy of $H_i$ in colour $i$, for some $1le ile k$. Let $phi_{k}(n,mathcal H)$ be the smallest number $phi$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $mathcal H$-decomposition with at most $phi$ elements. Extending the previous results of Liu and Sousa [Monochromatic $K_r$-decompositions of graphs, Journal of Graph Theory}, 76:89--100, 2014], we solve this problem when each graph in $mathcal H$ is a clique and $nge n_0(mathcal H)$ is sufficiently large.