Abstract
AbstractErdős, Gyárfás and Pyber showed that every r-edge-coloured complete graph Kn can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = \Omega(n^{-1/(2r)})$ , then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.
Highlights
An active line of current research concerns sparse random analogues of combinatorial theorems
An early example of this type of result was given by Rödl and Rucinski [24], who proved a random analogue of Ramsey’s theorem
Similar results have been obtained for asymmetric and hypergraph Ramsey problems
Summary
An active line of current research concerns sparse random analogues of combinatorial theorems. Given an edge-coloured graph, how many vertex-disjoint monochromatic cycles are necessary to cover its vertices?
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