Monochromatic Clique Decompositions of Graphs

Let $G$ be a graph whose edges are coloured with $k$ colours, and $\mathcal H=(H_1,\dots , H_k)$bea$k$-tupleofgraphs\mathrm{.}Amonochromatic$\mathcal H$-decompositionof$G$isapartitionoftheedgesetof$G$ such that each part is either a single edge or forms a monochromatic copy of $H_i$ in colour $i$, for some $1\le i\le k$. Let ${\varphi }_k(n,\mathcal{H})$ be the smallest number $\varphi$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $\mathcal{H}$-decomposition with at most $\varphi$ 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 $n\ge n_0(\mathcal{H})$ is sufficiently large.