Richard H. Hammack

On Cartesian skeletons of graphs

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 prime factorization property, and there are polynomial algorithms to determine the prime factors. This is most easily proved for the Cartesian product. For the other products, current proofs involve a notion of a Cartesian skeleton which transfers their multiplication properties to the Cartesian product. The present article introduces simplified definitions of Cartesian skeletons for the direct and strong products, and provides new, fast and transparent algorithms for their construction. Since the complexity of the prime factorization of the direct and the strong product is determined by the complexity of the the construction of the Cartesian skeleton, the new algorithms also improve the complexity of the prime factorizations of graphs with respect to the direct and the strong product. We indicate how these simplifications fit into the existing literature.

Cyclicity of graphs

A main result proved in this paper is the following. Theorem. Let G be a noncomplete graph on n vertices with degree sequence d1 ≥ d2 ≥ · · · ≥ dn and t ≥ 2 be a prime. Let m = gcd{t, di - dj: 1 ≤ i < j ≤ n} and set

Direct Product Factorization of Bipartite Graphs with Bipartition-reversing Involutions

d =\cases{1\ \ \ if\ m = t\ and\ \ m \not\mid\ d_{i}\ for\ 1 \leq i \leq n \cr 0\ \ \ otherwise.}

On uniqueness of prime bipartite factors of graphs

Then R(tG, ℤt) = t(n + d) - d, where R is the zero-sum Ramsey number. This settles, almost completely, problems raised in [Bialostocki & Dierker, J Graph Theory, 1994; Y. Caro, J Graph Theory, 1991].

Proof of a conjecture concerning the direct product of bipartite graphs

This paper presents a construction of a minimum cycle basis for the direct product of two complete graphs on three or more vertices. With the exception of two special cases, such bases consist entirely of triangles.

Centers of n-fold tensor products of graphs

Given a connected bipartite graph

Fast recognition of direct and strong products

G

Cancellation of direct products of digraphs

, we describe a procedure which enumerates and computes all graphs