tailieunhanh - Báo cáo toán học: "Distinguishing infinite graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Distinguishing infinite graphs. | Distinguishing infinite graphs Wilfried Imrich Montanuniversităt Leoben A-8700 Leoben Austria Sandi Klavzar Department of Mathematics and Computer Science FNM University of Maribor Gosposvetska cesta 84 2000 Maribor Slovenia Vladimir Trofimovy Institute of Mathematics and Mechanics Russian Academy of Sciences S. Kovalevskoy 16 620219 Ekaterinburg Russia trofimov@ Submitted Dec 9 2006 Accepted May 2 2007 Published May 11 2007 Mathematics Subject Classifications 05C25 05C80 03E10 Abstract The distinguishing number D G of a graph G is the least cardinal number @ such that G has a labeling with @ labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two that tree-like graphs with not more than continuum many vertices have distinguishing number two and determine the distinguishing number of many classes of infinite Cartesian products. For instance D Qn 2 where Qn is the infinite hypercube of dimension n. 1 Introduction The distinguishing number is a symmetry related graph invariant that was introduced by Albertson and Collins 2 and extensively studied afterward. In the last couple of Supported in part by the Ministry of Science of Slovenia under the grants P1-0297 and BI-AT 07-08-011 ySupported in part by the Russian Foundation for Basic Research under the grant 06-01-00378. The work was done in part during the visit of Montanuniversitat Leoben Leoben Austria in May 2006. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R36 1 years an amazing number of papers have been written on the topic. Let us mention just those that are directly related to our paper It was proved independently in 6 and 12 that for a finite connected graph G D G A 1 where A is the largest degree of G with equality if and only if G is a complete graph a regular complete bipartite graph or C5. For the computation of the distinguishing numbers of finite trees see 5