Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Distinguishing Number of Countable Homogeneous Relational Structures. | Distinguishing Number of Countable Homogeneous Relational Structures C. Laflamme University of Calgary Department of Mathematics and Statistics 2500 University Dr. NW. Calgary Alberta Canada T2N1N4 laf@ L. Nguyen Van The t N. SauerỈ Universite Paul Cezanne - Aix-Marseille III Department of Mathematics and Statistics Avenue de l escadrille Normandie-Niemen The University of Calgary Calgary 13397 Marseille Cedex 20 France Alberta Canada T2N1N4 Ft. nsauer@ lionel@ J Submitted Apr 18 2008 Accepted Jan 20 2010 Published Jan 29 2010 Mathematics Subject Classification 05E18 05C55 05C15 Abstract The distinguishing number of a graph G is the smallest positive integer r such that G has a labeling of its vertices with r labels for which there is no non-trivial automorphism of G preserving these labels. In early work Michael Albertson and Karen Collins computed the distinguishing number for various finite graphs and more recently Wilfried Imrich Sandi KlavZar and Vladimir Trofimov computed the distinguishing number of some infinite graphs showing in particular that the Random Graph has distinguishing number 2. We compute the distinguishing number of various other finite and countable homogeneous structures including undirected and directed graphs and posets. We show that this number is in most cases two or infinite and besides a few exceptions conjecture that this is so for all primitive homogeneous countable structures. Supported by NSERC of Canada Grant 690404 iThe author would like to thank the support of the Department of Mathematics Statistics Postdoctoral Program at the University of Calgary Supported by NSERC of Canada Grant 691325 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R20 1 1 Introduction The distinguishing number of a graph G was introduced in 1 by Michael Albertson and Karen Collins. It is the smallest positive integer r such that G has a labeling of its vertices into r labels for which there are no .