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: Lower bounds for identifying codes in some infinite grids. | Lower bounds for identifying codes in some infinite grids Ryan Martin Brendon Stanton Department of Mathematics Iowa State University Ames IA 50010 Submitted Apr 20 2010 Accepted Aug 27 2010 Published Sep 13 2010 Mathematics Subject Classification 05C70 68R10 94B65 Abstract An r-identifying code on a graph G is a set C c V G such that for every vertex in V G the intersection of the radius-r closed neighborhood with C is nonempty and unique. On a finite graph the density of a code is C V G which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of r in both the square and hexagonal grids. 1 Introduction Given a connected undirected graph G V E we define Br v -called the ball of radius r centered at v to be Br v u G V G d u v r . We call any nonempty subset C of V G a code and its elements codewords. A code C is called r-identifying if it has the properties 1. Br v n C 0 2. Br u n C Br v n C for all u v When C is understood we define Ir v Ir v C Br v n C. We call Ir v the identifying set of v. Vertex identifying codes were introduced in 6 as a way to help with fault diagnosis in multiprocessor computer systems. Codes have been studied in many graphs but of Research supported in part by NSA grant H98230-08-1-0015 and NSF grant DMS 0901008 and by an Iowa State University Faculty Professional Development grant. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R122 1 particular interest are codes in the infinite triangular square and hexagonal lattices as well as the square lattice with diagonals king grid . For each of these graphs there is a characterization so that the vertex set is Z X Z. Let Qm denote the set of vertices x y G Z X Z with x m and y m. We may then define the density of a code C by C n Qm D C limsup . m Qm Our first two theorems Theorem 1 and Theorem 2 rely on a key lemma Lemma 6 which gives a lower bound for the density of an .
