Local colourings and monochromatic partitions in complete bipartite graphs

Abstract

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.

18 pages

References in corpus (2)

• Partitioning 2-edge-colored graphs by monochromatic paths and cycles
• Monochromatic cycle partitions in local edge colourings

Cited by in corpus (1)

• Vertex covers by monochromatic pieces - A survey of results and problems