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Báo cáo toán học: " Vizing-like conjecture for the upper domination of Cartesian products of graphs – the proof"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Vizing-like conjecture for the upper domination of Cartesian products of graphs – the proof. | Vizing-like conjecture for the upper domination of Cartesian products of graphs - the proof Bostjan Bresar University of Maribor FEECS Smetanova 17 2000 Maribor Slovenia bostjan.bresar@uni-mb.si Submitted Jul 7 2004 Accepted Jun 30 2005 Published Jul 19 2005 Mathematics Subject Classifications 05C69 05C99 Abstract In this note we prove the following conjecture of Nowakowski and Rall For arbitrary graphs G and H the upper domination number of the Cartesian product G H is at least the product of their upper domination numbers in symbols r G H r G r H . A conjecture posed by Vizing 7 in 1968 claims that Vizing s conjecture For any graphs G and H Y G H y G y H where Y as usual denotes the domination number of a graph and G H is the Cartesian product of graphs G and H. It became one of the main problems of graph domination cf. surveys 2 and 4 Section 8.6 and two recent papers 1 6 . The unability of proving or disproving it lead authors to pose different variations of the original problem. Several such variations were studied by Nowakowski and Rall in the paper 5 from 1996. In particular they proposed the following Conjecture Nowakowski Rail For any graphs G and H r G H r G r H where r denotes the upper domination of a graph. In this note we prove this conjecture. In fact if both graphs G and H are nontrivial i.e. have at least two vertices we prove the following slightly stronger bound r G H r G r H 1. Supported by the Ministry of Education Science and Sport of Slovenia under the grant Z1-3073-0101-01. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N12 1 We start with basic definitions. For graphs G and H the Cartesian product G H is the graph with vertex set V G X V H where two vertices u vj and u2 v2 are adjacent if and only if either u1 u2 and v1v2 2 E H or v1 v2 and u1u2 2 E G . For a set of vertices S Q V G X V H let pG S pH S denote the natural projections of S to V G and V H respectively. A set S c V G of vertices in a graph G is called dominating if for every .