| Full text | |
| Author(s): |
Total Authors: 4
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| Affiliation: | [1] Univ Sao Paulo, Inst Matemat & Estat, BR-05508900 Sao Paulo - Brazil
[2] Emory Univ, Dept Math & Comp Sci, Atlanta, GA 30322 - USA
[3] Humboldt Univ, Inst Informat, D-10099 Berlin - Germany
[4] Rutgers State Univ, Dept Comp Sci, Piscataway, NJ 08854 - USA
Total Affiliations: 4
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| Document type: | Journal article |
| Source: | ADVANCES IN MATHEMATICS; v. 226, n. 6, p. 5041-5065, APR 1 2011. |
| Web of Science Citations: | 14 |
| Abstract | |
In 1983, Chvatal, Trotter and the two senior authors proved that for any Delta there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph K(N) with N >= Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Delta. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N(2-1/Delta)log(1/Delta)N) edges, with N = {[}B'n] for some constant B' that depends only on Delta. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Delta is O(n(2-1/Delta)log(1/Delta)n) Our approach is based on random graphs; in fact, we show that the classical Erdos-Renyi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Delta. The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed. (C) 2011 Elsevier Inc. All rights reserved. (AU) | |
| FAPESP's process: | 03/09925-5 - Foundations of computer science: combinatory algorithms and discrete structures |
| Grantee: | Yoshiharu Kohayakawa |
| Support Opportunities: | PRONEX Research - Thematic Grants |