Bidiakis cube Bidiakis cube In the mathematical field of graph theory, the Bidiakis cube is a 3-regular graph with 12 vertices and 18 edges. The Bidiakis cube is a cubic Hamiltonian graph and can be defined by the LCF notation [-6,4,-4]. The Bidiakis cube can also be constructed from a cube by adding edges across the top and bottom faces which connect the centres of opposite sides of the faces. The two additional edges need to be perpendicular to each other. With this construction, the Bidiakis cube is a polyhedral graph, and can be realized as a convex polyhedron. Therefore, by Steinitz's
Folded cube graph as many vertices, by identifying together (or contracting) every opposite pair of vertices. An order-"k" folded cube graph is "k"-regular with 2 vertices and 2"k" edges. The chromatic number of the order-"k" folded cube graph is two when "k" is even (that is, in this case, the graph is bipartite) and four when "k" is odd. The odd girth of a folded cube of odd order is "k", so for odd "k" greater than three the folded cube graphs provide a class of triangle-free graphs with chromatic number four and arbitrarily large odd girth. As a distance-regular graph with odd