Hypercubes As Direct Products

Direct Product Factorization of Bipartite Graphs with Bipartition-reversing Involutions

Given a connected bipartite graph G, we describe a procedure which enumerates and computes all graphs H (if any) for which there is a direct product factorization G ∼= H × K2. We apply this technique to the problems of factoring even cycles and hypercubes over the direct product. In the case of hypercubes, our work expands some known results by Brešar, Imrich, Klavžar, Rall, and Zmazek [Finite ...

Finite and infinite hypercubes as direct products

Nonfactorizable nonsingular hypercubes

We investigate nonsingular hypercubes, and prove several results, stating a condition for a hypercube to be the product of hypercubes of smaller dimensions. There is a shortage of higher dimensional nonsingular hypercubes in the literature. However, we show that the product of two nonsingular hypercubes is always nonsingular. Then we show how to construct four-dimensional nonsingular hypercubes...

Vertex Distinguishing Edge- and Total-Colorings of Cartesian and other Product Graphs

This paper studies edgeand total-colorings of graphs in which (all or only adjacent) vertices are distinguished by their sets of colors. We provide bounds for the minimum number of colors needed for such colorings for the Cartesian product of graphs along with exact results for generalized hypercubes. We also present general bounds for the direct, strong and lexicographic products.

On density of subgraphs of Cartesian products

In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs G of Cartesian products G1 × · · · × Gm of arbitrary connected graphs. Namely, we show that |E(G)| |V (G)| ≤ d2 max{dens(G1), . . . , dens(Gm)}e log |V (G)|, where dens(H) is the maximum ratio |E(H′)| |V (H′)| over all subgraphs H ′ of H. We introduce the notions of VC-dimension VC-dim(G) an...