Towards an edge-coloured Corrádi--Hajnal theorem

A classical result of Corrádi and Hajnal states that every graph $G$ on $n$ vertices with $n\in 3\mathbb{N}$ and $\delta (G)\ge 2n\mathrm{/}3$ contains a perfect triangle-tiling, i.e.,\ a spanning set of vertex-disjoint triangles. We explore a generalisation of this result to edge-coloured graphs. Let $G$ be an edge-coloured graph on $n$ vertices. The minimum colour degree ${\delta }^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v\in V(G)$, there are at least $k$ distinct colours on edges incident to $v$. We show that if ${\delta }^c(G)\ge (5\mathrm{/}6+\epsilon )n$, then $G$ has a spanning set of vertex-disjoint rainbow triangles. On the other hand, we find an example showing the bound should be at least $5n\mathrm{/}7$. We also discuss a related tiling problems on digraphs, which may be of independent interest.