The notion of a fractional Cartesian product and a subsequent notion of combinatorial dimension appeared first in a harmonic-analytic context in the course of filling ”analytic” gaps between successive (ordinary) Cartesian products of spectral sets [1, 2]. (Detailed accounts of this, and much more, appear in [4].) Combinatorial dimension is an index of interdependence: attached to a subset of a...

Let γt(G) and γpr(G) denote the total domination and the paired domination numbers of graph G, respectively, and let G ¤ H denote the Cartesian product of graphs G and H. In this paper, we show that γt(G)γt(H) ≤ 5γt(G ¤ H), which improves the known result γt(G)γt(H) ≤ 6γt(G ¤ H) given by Henning and Rall.

We determine the combinatorial discrepancy of the hypergraph H of cartesian products of d arithmetic progressions in the [N ]d–lattice ([N ] = {0, 1, . . . ,N − 1}). The study of such higher dimensional arithmetic progressions is motivated by a multi-dimensional version of van der Waerden’s theorem, namely the Gallai-theorem (1933). We solve the discrepancy problem for d–dimensional arithmetic ...

The study of domination in Cartesian products has received its main motivation from attempts to settle a conjecture made by V.G. Vizing in 1968. He conjectured that γ(G)γ(H) is a lower bound for the domination number of the Cartesian product of any two graphs G and H. Most of the progress on settling this conjecture has been limited to verifying the conjectured lower bound if one of the graphs ...

The k-fold Cartesian product of a graph G is defined as a graph on tuples (x1, . . . , xk) where two tuples are connected if they form an edge in one of the positions and are equal in the rest. Starting with G as a single edge gives G as a k-dimensional hypercube. We study the distributions of edges crossed by a cut in G across the copies of G in different positions. This is a generalization of...

We classify all factors of the Cartesian product of any two nonsingular type III , 0 < < 1, or II1, Chacon transformations, as well as the centralizer of nite Cartesian products of such transformations. AMS 1991 subject classi cation: 28D99.

In this work we investigate the behavior of various geodesic convexity parameters with respect to the Cartesian product operation for graphs. First, we show that the convex sets arising from geodesic convexity in a Cartesian product of graphs are exactly the same as the convex sets arising from the usual binary operation ⊕ for making a convexity space out of the Cartesian product of any two con...

Recall these definitions (from [2]): Definition (p. 116). In a graph G, a set S ⊆ V (G) is a dominating set if every vertex not in S has a neighbor in S. The domination number γ (G) is the minimum size of a dominating set in G. Definition (p. 193). The cartesian product of G and H, written G H, is the graph with vertex set V (G) × V (H) specified by putting (u, v) adjacent to (u′, v′) if and on...

The paper deals with Lukasiewicz filters. It analyzes some of their possible constructions and their basic properties. The Cartesian product of Lukasiewicz filters is introduced.