Non-disjoint monochromatic cycle covering conjecture

Let $K_n$ be a complete graph whose edges are coloured with $r$ colours. A cycle covering is a collection of $r$ monochromatic cycles whose union of vertex sets contains all vertices of $K_n$; the cycles need not be vertex-disjoint. Non-disjoint cycle covering conjecture. Suppose that the edges of $K_n$ are coloured with $r$ colours. There are $r$ not necessarily disjoint monochromatic cycles covering all the vertices in $K_n$. This weakens the refuted Erdős–Gyárfás–Pyber conjecture by removing disjointness; the source notes that the corresponding path statement is known for three colours, while the cycle conjecture remains a possible direction for further work.