Elsevier

Advances in Mathematics

Volume 226, Issue 6, 1 April 2011, Pages 5041-5065
Advances in Mathematics

Sparse partition universal graphs for graphs of bounded degree

Dedicated to the memory of Professor Richard H. Schelp
https://doi.org/10.1016/j.aim.2011.01.004Get rights and content
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Abstract

In 1983, Chvátal, Trotter and the two senior authors proved that for any Δ there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph KN with NBn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Δ. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N21/Δlog1/ΔN) edges, with N=Bn for some constant B that depends only on Δ. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Δ is O(n21/Δlog1/Δn). Our approach is based on random graphs; in fact, we show that the classical Erdős–Rényi 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 Δ.

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.

Keywords

Size-Ramsey numbers
Universal graphs
Regularity lemma
Random graphs
Inheritance of regularity

Cited by (0)

The collaboration between the first and the third authors was supported by a CAPES–DAAD collaboration grant.

1

The author was partially supported by FAPESP and CNPq through a Temático-ProNEx project (Proc. FAPESP 2003/09925-5) and by CNPq (Proc. 308509/2007-2, 485671/2007-7, 486124/2007-0 and 484154/2010-9).

2

The author was supported by NSF grants DMS 0300529 and DMS 0800070.

3

Current address: Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany.

4

The author was supported by NSF grants DMS 0100784 and DMS 0603745.