8,899 research outputs found

    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

    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

    Ramsey numbers for partially-ordered sets

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    We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl

    Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

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    In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

    Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

    Get PDF
    In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

    Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

    Get PDF
    In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
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