On MAXCUT in strictly supercritical random graphs, and coloring of random graphs and random tournaments

Lior Gishboliner, Michael Krivelevich, Gal Kronenberg

Research output: Contribution to journalArticlepeer-review

Abstract

We use a theorem by Ding, Lubetzky, and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of G~G(n, 1+ε/n) in terms of ɛ. We then apply this result to prove the following conjecture by Frieze and Pegden. For every ε>0, there exists lε such that w.h.p. G~G(n, 1+ε/n) is not homomorphic to the cycle on 2lε+1 vertices. We also consider the coloring properties of biased random tournaments. A p-random tournament on n vertices is obtained from the transitive tournament by reversing each edge independently with probability p. We show that for p = Θ( 1/n ) the chromatic number of a p-random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case P=1+ε/n we use the aforementioned result on MAXCUT and show that in fact w.h.p. one needs to reverse Θ(ε3)n edges to make it 2-colorable.

Original languageEnglish
Pages (from-to)545-559
Number of pages15
JournalRandom Structures and Algorithms
Volume52
Issue number4
DOIs
StatePublished - Jul 2018

Keywords

  • chromatic number
  • graph coloring
  • max-cut
  • random graph
  • random tournament

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