Partitioning 2-edge-coloured bipartite graphs into monochromatic cycles
Given an $r$-edge-colouring of the edges of a graph $G$, we say that it can be partitioned into $p$ monochromatic cycles when there exists a set of $p$ vertex-disjoint monochromatic cycles covering all the vertices of $G$. In the literature of this problem, an edge and a single vertex both count as a cycle. We show that for every $2$-colouring of the edges of a complete balanced bipartite graph, $K_{n,n}$, it can be partitioned into at most 4 monochromatic cycles. This type of question was first studied in 1970 for complete graphs and in 1983, by Gyárfás and Lehel, for $K_{n,n}$. In 2014, Pokrovskiy showed for all $n$ that, given any $2$-colouring of its edges, $K_{n,n}$ can be partitioned into at most three monochromatic paths. It turns out that finding monochromatic cycles instead of paths is a natural question that has also been raised for other graphs. In 2015, Schaudt and Stein showed that 14 cycles are sufficient for sufficiently large $2$-edge-coloured $K_{n,n}$.