Towards an edge-coloured Corr\'adi--Hajnal theorem

A classical result of Corr\'adi and Hajnal states that every graph G on n vertices with n∈ 3N and δ(G) ≥ 2n/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 δᶜ(G) of G is the largest integer k such that, for every vertex v ∈ V(G), there are at least k distinct colours on edges incident to v. We show that if δᶜ(G) ≥ (5/6 + ε) 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 $5n/7$. We also discuss a related tiling problems on digraphs, which may be of independent interest.