tailieunhanh - Báo cáo toán học: "General bounds for identifying codes in some infinite regular graphs"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: General bounds for identifying codes in some infinite regular graphs. | General bounds for identifying codes in some infinite regular graphs Irene Charon CNRS ENST 46 rue Barrault 75634 Paris Cedex 13 - France charon@ Olivier Hudry CNRS ENST 46 rue Barrault 75634 Paris Cedex 13 - France hudry@ Iiro Honkala University of Turku Department of Mathematics 20014 Turku Finland honkala@ Antoine Lobstein CNRS ENST 46 rue Barrault 75634 Paris Cedex 13 - France lobstein@ Submitted October 10 2000 Accepted November 14 2001. MR Subject Classifications 05C70 68R10 94B65 Abstract Consider a connected undirected graph G V E and a subset of vertices C. If for all vertices v 2 V the sets Br v n C are all nonempty and pairwise distinct where Br v denotes the set of all points within distance r from v then we call C an r-identifying code. We give general lower and upper bounds on the best possible density of r-identifying codes in three infinite regular graphs. 1 Introduction Let G V E be a connected undirected graph finite or infinite we define Br v the ball of radius r centred at a vertex v 2 V by Br v x 2 V d X v rg where d X v denotes the number of edges in any shortest path between v and X. Whenever d X v r we say that X and v r-cover each other or simply cover if there is no ambiguity . A set of vertices covers a vertex if at least one of its elements does. Research supported by the Academy of Finland Grant 44002. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R39 1 We call any nonempty subset C of V a code and its elements codewords. A code C is called r-identifying or identifying if the sets Br v n C v 2 V are all nonempty and pairwise distinct. The set Br v n C is called the r-identifying set or identifying set of v and will be denoted by I v . Two vertices which have different identifying sets are said to be r-separated or separated. The concept of identifying code was introduced in 14 . It was further studied for different types of graphs . in 1 13 . In this paper we will study the following .