The uniqueness of the cubic lattice graph

Abstract A cubic lattice graph is defined as a graph G, whose vertices are the ordered triplets on n symbols, such that two vertices are adjacent if and only if they have two coordinates in common. Laskar characterized these graphs for n>7 by means of five conditions. In this paper the same characterization is shown to hold for all n except for n=4, where the existence of exactly one exceptional case is demonstrated.