Discrete Mathematics 276 (2004) 59–64 www.elsevier.com/locate/disc Dierence graphs  Endre Boros a , Vladimir Gurvich a , Roy Meshulam b a RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854-8003, USA b Mathematics Department of the Technion, Israel Institute of Technology, Haifa, Israel Abstract Intersection and measured intersection graphs are quite common in the literature. In this paper we introduce the analogous concept of measured dierence graphs: Given an arbitrary hypergraph H = {H1;:::;Hn}, let us associate to it a graph on vertex set [n]= {1; 2;:::;n} in which (i; j) is an edge i the corresponding sets Hi and Hj are “suciently dierent”. More precisely, given an integer threshold k , we consider three denitions, according to which (i; j) is an edge i (1) |Hi \ Hj | +|Hj \ Hi | ¿ 2k , (2) max{|Hi \ Hj |; |Hj \ Hi |} ¿ k , and (3) min{|Hi \ Hj |; |Hj \ Hi |} ¿ k . It is not dicult to see that each of the above denes hereditary graph classes, which are monotone with respect to k . We show that for every graph G there exists a large enough k such that G arises with any of the denitions above. We prove that with the rst two denitions one may need k = (log n) in any such realizations of certain graphs on n vertices. However, we do not know a graph G which could not be realized by the last denition with k =2. c  2003 Elsevier B.V. All rights reserved. Keywords: Graph realization; Intersection graphs; Dierence graphs 1. Introduction The intersection and measured intersection graphs are quite often considered in the literature; see for instance the surveys [3, chapter 4]; [5,6]. Here we introduce an analogous concept of measured dierence graphs. Let us introduce the notation [n]= {1; 2;:::;n}. For a set X and integer l let us denote by ( X l ) the family of all subsets of X of size l.  The authors gratefully acknowledge the partial support by the National Science Foundation (Grant IIS-0118635), the Oce of Naval Research (Grants N00014-92-J-1375), and by DIMACS, the National Science Foundation Center for Discrete Mathematics and Theoretical Computer Science. E-mail addresses: boros@rutcor.rutgers.edu (E. Boros), gurvich@rutcor.rutgers.edu (V. Gurvich), meshulam@tx.technion.ac.il (R. Meshulam). 0012-365X/$-see front matter c  2003 Elsevier B.V. All rights reserved. doi:10.1016/S0012-365X(03)00321-2