This paper is concerned with the bipanconnectivity of Cartesian product networks. It provides sufficient conditions for a Cartesian product network to be bipanconnected and uses this general result to re-prove existing results as regards the k-ary n-cube, due to Stewart and Xiang and to Hsieh and Lin, as well as to extend these results to multidimensional toroidal meshes.

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...

We are learning strange things about cartesian products. Shapiro has shown that the cartesian product of a line and a certain example described by Whitehead [ló] is topologically E4. The example is a bounded open subset of E3 that is simply connected but which contains a simple closed curve which does not lie in any topological cube in it [3]. Glimm has given a proof of this result in [9] and a...

The combinatorial dimension relative to an arbitrary fractional Cartesian product is defined. Relations between dimensions in certain archetypal instances are derived. Random sets with arbitrarily prescribed dimensions are produced; in particular, scales of combinatorial dimension are shown to be continuously and independently calibrated. A combinatorial concept of cylindricity is key. © 2004 W...