Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, September 29, 2016, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Nemanja Skoric

Covering sparse graphs by monochromatic cycles

In a classical paper, Erdos, Gyarfas and Pyber prove that there exists a function pc : N -> N such that for any r-coloring of the edge set of the complete graph K_n there is a covering (even partitioning) of the vertex set by at most pc(r) many monochromatic cycles. In particular, this means that if the number of colors is constant only constantly many monochromatic cycles are needed to cover K_n. Since the original paper there have been many improvements on the value of pc(r) and different generalizations of the problem. In this paper we initiate the study of the problem for the case of random graphs. We prove that for any constant eps > 0 and p >= n^{-1/2 + eps} w.h.p. G ̴ Gnp has the following property: for any 2-coloring of G the graph can be covered by constantly many monochromatic cycles. The bound on p can not be much improved as when p is significantly less than (log n / n)^{1/2} the number of monochromatic cycles needed to cover Gnp grows to infinity with n.
Joint work with Daniel Korandi, Frank Mousset, Rajko Nenadov and Benjamin Sudakov.

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