The distinguishing number of the hypercube Original Research Article

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 Hd, that D(Hd)=3 if d∈{2,3} and D(Hd)=2 if d⩾4. It is also shown that D(Hd2)=4 for d∈{2,3} and D(Hd2)=2 for d⩾4, where Hd2 denotes the square of the d-dimensional hypercube. This solves the distinguishing number for hypercubic graphs and their squares.