### The determining number of Kneser graphs

*José Cáceres, Delia Garijo, Antonio González, Alberto Márquez, María Luz Puertas*

#### 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 K_{n: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 S_{n}acting on the k-subsets of {1,…, n}. Then, we establish for which Kneser graphs K_{n: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.Full Text: PDF PostScript