Distinguishing labellings of group action on vector spaces and graphs

Abstract

Suppose Γ is a group acting on a set X. A k-labeling of X is a mapping c : X → { 1 , 2 , … , k } . A labeling c of X is distinguishing (with respect to the action of Γ) if for any g ∈ Γ , g ≠ id X , there exists an element x ∈ X such that c ( x ) ≠ c ( g ( x ) ) . The distinguishing number, D Γ ( X ) , of the action of Γ on X is the minimum k for which there is a k-labeling which is distinguishing. This paper studies the distinguishing number of the linear group GL n ( K ) over a field K acting on the vector space K n and the distinguishing number of the automorphism group Aut ( G ) of a graph G acting on V ( G ) . The latter is called the distinguishing number of the graph G and is denoted by D ( G ) . We determine the value of D GL n ( K ) ( K n ) for all fields K and integers n. For the distinguishing number of graphs, we study the possible value of the distinguishing number of a graph in terms of its automorphism group, its maximum degree, and other structure properties. It is proved that if Aut ( G ) = S n and each orbit of Aut ( G ) has size less than ( n 2 ) , then D ( G ) = ⌈ n 1 / k ⌉ for some positive integer k. A Brooks type theorem for the distinguishing number is obtained: for any graph G, D ( G ) ⩽ Δ ( G ) , unless G is a complete graph, regular complete bipartite graph, or C 5 . We introduce the notion of uniquely distinguishable graphs and study the distinguishing number of disconnected graphs.