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
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.
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