Partitioning Random Graphs Into Monochromatic Components

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Erdős, Gyàrfàs, and Pyber (1991) conjectured that every r-colored complete graph can be partitioned into at most r – 1 monochromatic components; this is a strengthening of a conjecture of Lovàsz (1975) and Ryser (1970) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into r monochromatic components is possible for sufficiently large r-colored complete graphs. We start by extending Haxell and Kohayakawa’s result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if (formula presented), then a.a.s. in every 2- coloring of G(n; p) there exists a partition into two monochromatic components, and for r ≥ 2 if p << (formula presented), then a.a.s. there exists an r-coloring of G(n, p) such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gyàrfàs (1977) about large monochromatic components in r-colored complete graphs. We show that if p = ω(1)/n, then a.a.s. in every r-coloring of G(n, p) there exists a monochromatic component of order at least (formula presented).

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