We show that for any $2$-local colouring of the edges of a complete bipartite graph, its vertices can be covered with at most $3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.
Almost partitioning a 3-edge-coloured $K_{n,n}$ into 5 monochromatic cycles
A tropical morphism related to the hyperplane arrangement of the complete bipartite graph
The extremal number of the subdivisions of the complete bipartite graph
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