Almost partitioning a 3-edge-coloured $K_{n,n}$ into 5 monochromatic cycles

TL;DR

This paper proves that in any 3-colouring of a complete bipartite graph, a small number of monochromatic cycles can cover almost all or all vertices, advancing understanding of monochromatic cycle decompositions.

Contribution

It establishes new bounds on the number of monochromatic cycles needed to cover nearly all or all vertices in 3-coloured bipartite graphs.

Findings

5 cycles cover all but o(n) vertices

18 cycles cover all vertices

Results improve bounds on monochromatic cycle coverings

Abstract

We show that for any colouring of the edges of the complete bipartite graph $K_{n,n}$ with 3 colours there are 5 disjoint monochromatic cycles which together cover all but $o(n)$ of the vertices. In the same situation, 18 disjoint monochromatic cycles together cover all vertices.