Using determining sets to distinguish Kneser graphs

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 ⊆ 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 K n:k with n ≥ 6 and k ≥ 2 is 2-distinguishable.