The construction of the halved cube graph can be reformulated in terms of binary numbers.

For instance, the halved cube graph of dimension four may be formed from an ordinary three-dimensional cube by keeping the cube edges and adding edges connecting pairs of vertices that are on opposite corners of the same square face.

As with the hypercube graphs, and their isometric (distance-preserving) subgraphs the partial cubes, a halved cube graph may be embedded isometrically into a real vector space with the Manhattan metric (L1 distance function).

The same is true for the isometric subgraphs of halved cube graphs, which may be recognized in polynomial time; this forms a key subroutine for an algorithm which tests whether a given graph may be embedded isometrically into a Manhattan metric.

For the graphs of dimension three and four, four colors are needed to eliminate all symmetries.

[5] The two graphs shown are symmetric Dn and Bn Petrie polygon projections (2(n − 1) and n dihedral symmetry) of the related polytope which can include overlapping edges and vertices.