Monochromatic partitions in 2-edge-coloured bipartite graphs
arxiv(2024)
摘要
We study two variations of the Gyarfas–Lehel conjecture on the minimum
number of monochromatic components needed to cover an edge-coloured complete
bipartite graph. Specifically, we show the following. - For p>> (logn/n)^1/2, w.h.p. every 2-colouring of the random bipartite graph G G(n,n,p)
admits a cover of all but O(1/p) vertices of G using at most three
vertex-disjoint monochromatic components. - For every 2-colouring of a
bipartite graph G with parts of size n and minimum degree (13/16+o(1))n, the
vertices of G can be covered using at most three vertex-disjoint monochromatic
components.
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