tailieunhanh - Báo cáo toán học: "Using Determining Sets to Distinguish Kneser Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Using Determining Sets to Distinguish Kneser Graphs. | Using Determining Sets to Distinguish Kneser Graphs Michael O. Albertson Debra L. Boutin Department of Mathematics and Statistics Smith College Northampton MA 01063 albertson@ Department of Mathematics Hamilton College Clinton NY 13323 dboutin@ Submitted Sep 10 2006 Accepted Jan 19 2007 Published Jan 29 2007 Mathematics Subject Classihcation 05C25 05C78 Abstract This work introduces the technique of using a carefully chosen determining set to prove the existence of a distinguishing labeling using few labels. A graph G is said to be d-distinguishable if there is a labeling of the vertex set using 1 . d so that no nontrivial automorphism of G preserves the labels. A set of vertices S c V G is a determining set for G if every automorphism of G is uniquely determined by its action on S. We prove that a graph is d-distinguishable if and only if it has a determining set that can be d 1 -distinguished. We use this to prove that every Kneser graph Kn k with n 6 and k 2 is 2-distinguishable. 1 Introduction The distinguishing number of a graph G is the smallest integer d so that each vertex of G can be labeled with an integer from 1 . dg in such a way that no automorphism of G other than the identity preserves the labels. Albertson and Collins introduced distinguishing in 3 . There has been a flurry of activity on distinguishing in the last few years see . 1 2 4 6 7 8 9 13 14 15 17 18 20 . A subset S of vertices of a graph G is called a determining set if whenever two automorphisms agree on the elements of S they agree on all of G. That is the image of S under an arbitrary automorphism determines the automorphism completely. The determining number of a graph G is the smallest integer r so that G has a determining set of size r. Boutin introduced determining in 5 . Both distinguishing labelings and determining sets illuminate and quantify the symmetry of a graph. However there are fundamental differences between these two concepts. A .