209 research outputs found
Ramsey games with giants
The classical result in the theory of random graphs, proved by Erdos and
Renyi in 1960, concerns the threshold for the appearance of the giant component
in the random graph process. We consider a variant of this problem, with a
Ramsey flavor. Now, each random edge that arrives in the sequence of rounds
must be colored with one of R colors. The goal can be either to create a giant
component in every color class, or alternatively, to avoid it in every color.
One can analyze the offline or online setting for this problem. In this paper,
we consider all these variants and provide nontrivial upper and lower bounds;
in certain cases (like online avoidance) the obtained bounds are asymptotically
tight.Comment: 29 pages; minor revision
Partitioning random graphs into monochromatic components
Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every -colored
complete graph can be partitioned into at most monochromatic components;
this is a strengthening of a conjecture of Lov\'asz (1975) 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 monochromatic
components is possible for sufficiently large -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 , then a.a.s. in every -coloring of there exists
a partition into two monochromatic components, and for if , then a.a.s. there exists an -coloring
of 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
Gy\'arf\'as (1977) about large monochromatic components in -colored complete
graphs. We show that if , then a.a.s. in every
-coloring of there exists a monochromatic component of order at
least .Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics
Volume 24, Issue 1 (2017) Paper #P1.1
Combinatorics, Probability and Computing
One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly effective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b
On the Power of Choice for k-Colorability of Random Graphs
In an r-choice Achlioptas process, random edges are generated r at a time, and an online strategy is used to select one of them for inclusion in a graph. We investigate the problem of whether such a selection strategy can shift the k-colorability transition; that is, the number of edges at which the graph goes from being k-colorable to non-k-colorable.
We show that, for k ? 9, two choices suffice to delay the k-colorability threshold, and that for every k ? 2, six choices suffice
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