tailieunhanh - Báo cáo toán học: "A characterization for sparse ε-regular pairs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A characterization for sparse ε-regular pairs. | A characterization for sparse -regular pairs Stefanie Gerke Angelika Steger Institute of Theoretical Computer Science ETH Zurich CH-8092 Zurich email sgerke@ steger@ Submitted Dec 9 2005 Accepted Dec 10 2006 Published Jan 3 2007 Mathematical Subject Classification 05C75 05C80 Abstract We are interested in e -regular bipartite graphs which are the central objects in the regularity lemma of Szemeredi for sparse graphs. A bipartite graph G A B E with density p E A B is e -regular if for all sets A c A and B c B ofsize A e A and B0 B it holds that eG A B0 A0 B -p ep. In this paper we prove a characterization for e -regularity. That is we give a set of properties that hold for each e -regular graph and conversely if the properties of this set hold for a bipartite graph then the graph is f e -regular for some appropriate function f with f e 0 as e 0. The properties of this set concern degrees of vertices and common degrees of vertices with sets of size 0 1 p where p is the density of the graph in question. 1 Introduction We are interested in -regular pairs which play a central role in the famous regularity lemma of Szemeredi 11 . In fact we consider a generalisation of the regularity concept that has been introduced by Kohayakawa and Rodl 7 . Following Kohayakawa and Rodl we say that a bipartite graph G A B E with density p E A B is ẽ -regular if for all sets A c A and B0 c of size A01 A and B we have eG A0 B0 A B p p 1 where eG Af Bf denotes the number of edges between A0 and B0 in G. In the original definition of -regularity by Szemeredi the p on the right-hand-side of 1 is not present. For the remainder we will use the notation -regular in contrast to -regular when referring to the original definition by Szemeredi. Note that as p 1 every -regular graph is also -regular. Vice versa this is not the case and in particular it is easily verified THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R4 1 that every bipartite graph with less than 3 A B I .