The determining number of Kneser graphs

Abstract

A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G is the minimum cardinality of a determining set of G. This paper studies the determining number of Kneser graphs. First, we compute the determining number of a wide range of Kneser graphs, concretely Kn:k with n ≥ k(k+1)/2 + 1. In the language of group theory, these computations provide exact values for the base size of the symmetric group Sn acting on the k-subsets of (1, ⋯ , n). Then, we establish for which Kneser graphs Kn:k the determining number is equal to n - k, answering a question posed by Boutin. Finally, we find all Kneser graphs with fixed determining number 5, extending the study developed by Boutin for determining number 2, 3 or 4. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.