The distinguishing number of the hypercube

Abstract

The distinguishing number of a graph G is the minimum number of colors for which there exists an assignment of colors to the vertices of G so that the group of color-preserving automorphisms of G consists only of the identity. It is shown, for the d -dimensional hypercubic graphs H d , that D ( H d )=3 if d ∈{2,3} and D ( H d )=2 if d ⩾4. It is also shown that D ( H d 2 )=4 for d ∈{2,3} and D ( H d 2 )=2 for d ⩾4, where H d 2 denotes the square of the d -dimensional hypercube. This solves the distinguishing number for hypercubic graphs and their squares.