Partitioning edge-coloured complete symmetric digraphs into monochromatic complete subgraphs

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

A conjecture of Erdős, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertexdisjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2. In this paper we show that in fact this conjecture is false for all r ≥ 3. In contrast to this, we show that in any edge-colouring of a complete g...

Partitioning two-coloured complete multipartite graphs into monochromatic paths

We show that any complete k-partite graph G on n vertices, with k ≥ 3, whose edges are two-coloured, can be covered by two vertex-disjoint monochromatic paths of distinct colours, 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, ...

Partitioning 3-coloured complete graphs into three monochromatic paths

On monochromatic subgraphs of edge-colored complete graphs

In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the monochromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that eve...

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

We show that any complete k-partite graph G on n vertices, with k ≥ 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