Báo cáo toán học: "A note on packing chromatic number of the square lattice"

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: A note on packing chromatic number of the square lattice. | A note on packing chromatic number of the square lattice Roman Soukal Premysl Holub Submitted Oct 27 2009 Accepted Mar 2 2010 Published Mar 15 2010 Mathematics Subject Classification 05C12 05C15 Abstract The concept of a packing colouring is related to a frequency assignment problem. The packing chromatic number XP G of a graph G is the smallest integer k such that the vertex set V G can be partitioned into disjoint classes Xi . Xk where vertices in Xi have pairwise distance greater than i. In this note we improve the upper bound on the packing chromatic number of the square lattice. Keywords Packing chromatic number Packing colouring Square lattice 1 Introduction In this paper we consider only simple undirected graphs. We use 1 for terminology and notation not defined here. Let distG x y denote the distance between vertices x and y in G. The Cartesian product GOH of graphs G and H is the graph with vertex set V G X V H where vertices x y and x yz are adjacent whenever xx G E G and y yt or x x and yy G E H . For two infinite paths P1 P2 the Cartesian product PfiJP2 is usually called the infinite square lattice. The concept of the packing colouring comes from the area of frequency planning in wireless networks and was introduced by Goddard et al. in 3 under the name broadcast colouring. The packing chromatic number of a graph G denoted by XP G is the smallest integer k such that V G can be partitioned into k disjoint sets X1 . Xk where for each pair of vertices x y G Xi distG x y i for every i 1 . k. In other words vertices with the same colour i are pairwise at distance greater than i. Department of computer science and engineering Univerzitni 22 306 14 Pilsen Czech Republic e-mail soukal@. Research supported by the Grant Agency of the Czech Republic - the project GA 201 09 0097. Department of Mathematics University of West Bohemia and Institute for Theoretical Computer Science ITI Charles University Univerzitni 22 306 14 Pilsen Czech Republic e-mail .