Partitioning two-coloured complete multipartite graphs into monochromatic paths and cycles

TL;DR

This paper proves that large two-coloured complete multipartite graphs can be partitioned into monochromatic paths and cycles, extending previous results and providing bounds on the number of cycles needed.

Contribution

It extends known results to complete multipartite graphs with specific size constraints, showing they can be covered with monochromatic paths and cycles.

Findings

Any such graph can be covered with two monochromatic paths of distinct colours.

Almost all vertices can be covered with two monochromatic cycles, excluding a small fraction.

The entire graph can be covered with at most 14 monochromatic cycles.

Abstract

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of $G$ contains at most $n/2$ vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but $o(n)$ vertices of the graph can be covered with two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded. From this we derive that the whole graph, if large enough, may be covered with 14 vertex-disjoint monochromatic cycles.