Monochromatic Clique Decompositions of Graphs

H., Liu, Pikhurko O., and Sousa Teresa. "Monochromatic Clique Decompositions of Graphs." Journal of Graph Theory. 80 (2015): 287-298.


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 \emph{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 $1\le i\le k$. 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", \emph{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.

general-mono-clique.pdf263.54 KB