tailieunhanh - Báo cáo toán học: "Matrix-free proof of a regularity characterization"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Matrix-free proof of a regularity characterization. | Matrix-free proof of a regularity characterization A. Czygrinow Department of Mathematics and Statistics Arizona State University Tempe Arizona 85287 USA andrzej@ B. Nagle Department of Mathematics and Statistics University of Nevada Reno Nevada 89557 USA nagle@ Submitted May 28 2003 Accepted Oct 7 2003 Published Oct 13 2003 MR Subject Classifications 05C35 05C80 Abstract The central concept in Szemeredi s powerful regularity lemma is the so-called e-regular pair. A useful statement of Alon et al. essentially equates the notion of an e-regular pair with degree uniformity of vertices and pairs of vertices. The known proof of this characterization uses a clever matrix argument. This paper gives a simple proof of the characterization without appealing to the matrix argument of Alon et al. We show the e-regular characterization follows from an application of Szemeredi s regularity lemma itself. 1 Introduction The well-known Szemeredi Regularity Lemma 7 cf. 4 or 5 may be the single most powerful tool in extremal graph theory. Roughly speaking this lemma asserts that every large enough graph may be decomposed into constantly many random-like induced bipartite subgraphs . e-regular pairs . A property of the e-regular pairs obtained from Szemeredi s lemma is studied in this note. Suppose G U u V E is a bipartite graph. For nonempty subsets U c U and V c V let G U V u v E E u E U v E V be the subgraph of G induced on U and V . Set d U V IG U V U -1 V -1 to be the density of U and V . For e 0 we say G U u V E is e-regular if for all U c U U s U and V c V V eV we have1 d U V d U V e. 1For simplicity of calculations in this paper s a b t is short for a b t s a b t. THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R39 1 Equivalent conditions for e-regularity We consider the following two conditions for a bipartite graph G U u V E with fixed density d where whenever needed we assume U and V are sufficiently large . For 0 e 5 1 consider G1 G1 e G is .