Bounded decomposition into rainbow powers of cycles

Prove that, for every fixed integer r≥2, every properly edge-coloured complete graph K_n can be decomposed into a bounded number of vertex-disjoint rainbow r-cycles, where an r-cycle is the r-th power of a cycle.

Background

The paper proves that every sufficiently large properly edge-coloured complete graph can be partitioned into two vertex-disjoint rainbow cycles. It then considers analogous decomposition problems for powers of cycles. For an integer r≥2, the r-th power of a cycle is defined as the graph in which two vertices are adjacent whenever their distance along the underlying cycle is at most r.

By Vizing’s Theorem, a properly edge-coloured K_n uses at most n colours. Consequently, any rainbow r-cycle in such a colouring can contain at most floor(n/r) vertices, so two rainbow r-cycles cannot generally cover all vertices. The authors therefore formulate the unresolved possibility that a bounded number of rainbow r-cycles may suffice, with the bound understood for fixed r.