Cubic Partial Cubes from Simplicial Arrangements

Classification of vertex-transitive cubic partial cubes

Partial cubes are graphs isometrically embeddable into hypercubes. In this paper it is proved that every cubic, vertex-transitive partial cube is isomorphic to one of the following graphs: K2ƒC2n, for some n ≥ 2, generalized Petersen graph G(10, 3), permutahedron, truncated cuboctahedron, or truncated icosidodecahedron. This complete classification of cubic, vertex-transitive partial cubes, a f...

Tribes of Cubic Partial Cubes

Partial cubes are graphs isometrically embeddable into hypercubes. Three infinite families and a few sporadic examples of cubic partial cubes are known. The concept of a tribe is introduced as means to systematize the known examples and establish relations among them. Efficient methods of computation of tribes are developed and several concrete tribes, that include known, as well as new cubic p...

On cubic and edge-critical isometric subgraphs of hypercubes

All cubic partial cubes (i.e., cubic isometric subgraphs of hypercubes) up to 30 vertices and all edge-critical partial cubes up to 14 vertices are presented. The lists of graphs were confirmed by computer search to be complete. Non-trivial cubic partial cubes on 36, 42, and 48 vertices are also constructed.

Supplementary XRD Theory

The structure of a crystalline material may be described by a crystal lattice structure and the locations of atoms in this lattice. The crystal lattice is a 3-dimensional periodic array that defines the symmetries, i.e. shape, of the crystal. There are seven possible crystal lattices: cubic, hexagonal, tetragonal, rombohedral, orthorhombic, monoclinic and triclinic. When we define the possible ...

Cubic inflation, mirror graphs, regular maps, and partial cubes