Under suitable conditions of connectivity or non-bipartiteness, each of the three standard graph products (the Cartesian product, the direct product and the strong product) satisfies the unique primeâ€¦ (More)

We construct a minimum cycle basis for the direct product G Ã— Cq where G is a connected non-bipartite graph, Cq is an odd cycle and GÃ—Cq is triangle-free. These bases are expressed in terms of theâ€¦ (More)

We define what appears to be a new construction. Given a graph G and a positive integer k, the reduced kth power of G, denoted G, is the configuration space in which k indistinguishable tokens areâ€¦ (More)

A circular cover of a graph G is a cover {X0, Â· Â· Â· , Xnâˆ’1} of the topological space G by closed connected subsets, indexed over Zn, with the following properties: Each element in the cover containsâ€¦ (More)

Given graphs A, B and C for which A Ã— C âˆ¼= B Ã— C, it is not generally true that A âˆ¼= B. However, it is known that A Ã— C âˆ¼= B Ã— C implies A âˆ¼= B provided that C is non-bipartite, or that there areâ€¦ (More)

This note describes fast algorithms for computing the prime factors of connected, nonbipartite graphs with respect to the direct product, and of connected graphs with respect to the strong product.â€¦ (More)

A fundamental result, due to Sabidussi and Vizing, states that every connected graph has a unique prime factorization relative to the Cartesian product; but disconnected graphs are not uniquely primeâ€¦ (More)

A perfect r-code in a graph is a subset of the graphâ€™s vertices with the property that each vertex in the graph is within distance r of exactly one vertex in the subset. We prove that the n-foldâ€¦ (More)

We define a new graph operation called the kth inner power. The constructionâ€”which is somewhat analogous to the kth power with respect to the direct productâ€”seems to lend itself nicely to certainâ€¦ (More)