1,195 research outputs found

    Local colourings and monochromatic partitions in complete bipartite graphs

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    We show that for any 22-local colouring of the edges of the balanced complete bipartite graph Kn,nK_{n,n}, its vertices can be covered with at most~33 disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite rr-locally coloured graph with O(r2)O(r^2) disjoint monochromatic cycles.\\ We also determine the 22-local bipartite Ramsey number of a path almost exactly: Every 22-local colouring of the edges of Kn,nK_{n,n} contains a monochromatic path on nn vertices.Comment: 18 page

    Large rainbow matchings in large graphs

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    A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex vv is the number of different colors on edges incident to vv. We show that if nn is large enough (namely, n≥4.25k2n\geq 4.25k^2), then each nn-vertex graph GG with minimum color degree at least kk contains a rainbow matching of size at least kk
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