The cost of 2-distinguishing hypercubes

Abstract

A graph G is said to be 2-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the labels. The minimum size of a label class, over all 2-distinguishing labelings, is called the cost of 2-distinguishing, denoted by ρ(G). For n≥4 the hypercubes Qn are 2-distinguishable, but the values for ρ(Qn) have been elusive, with only bounds and partial results previously known. This paper settles the question. The main result can be summarized as: for n≥4, ρ(Qn)∈{1+⌈log2⁡n⌉,2+⌈log2⁡n⌉}. Exact values are found using a recursive relationship involving a new parameter νm, the smallest integer for which ρ(Qνm)=m. The main result is4≤n≤12⟹ρ(Qn)=5, and 5≤m≤11⟹νm=4; for m≥6,ρ(Qn)=m⇔2m−2−νm−1+1≤n≤2m−1−νm; for n≥5,νm=n⇔2n−1−ρ(Qn−1)+1≤m≤2n−ρ(Qn).