Abstract
Graph Theory 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.
Highlights
The determining number of a graph G = (V (G), E(G)) is the minimum cardinality of a set S ⊆ V (G) such that the automorphism group of the graph obtained from G by fixing every vertex in S is trivial
We continue the study on the determining number of Kneser graphs carried out by Boutin (2006), introducing a different technique from the tools used in that article
The main tools used in Boutin (2006) to find determining sets or to bound the determining number of Kneser graphs are based on characteristic matrices and vertex orbits
Summary
The main tools used in Boutin (2006) to find determining sets or to bound the determining number of Kneser graphs are based on characteristic matrices and vertex orbits. We want to stress that since the automorphism group of the Kneser graph Kn:k is the action of the symmetric group Sn on the k-subsets of [n] (see Godsil and Royle (2001)), our results in Section 2 provide exact values for the base size of Sn acting on the k-subsets of [n] for n. Our second main result concerns the question of whether there exists an infinite family of Kneser graphs Kn:k with k ≥ 2 and determining number n − k, which was posed by Boutin (2006). We conclude the paper with some remarks and open problems
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