On the Orchard crossing number of complete bipartite graphs

Let G be an abstract graph. Motivated by the Orchard relation, introduced in [3, 4], we have defined the Orchard crossing number of G [5], in a similar way to the well-known rectilinear crossing number of an abstract graph G (denoted by cr(G), see [1, 8]). A general reference for crossing numbers can be [6]. The Orchard crossing number is interesting for several reasons. First, it is based on the Orchard relation which is an equivalence relation on the vertices of a graph, with at most two equivalence classes (see [3]). Moreover, since the Orchard relation can be defined for higher dimensions too (see [3]), hence the Orchard crossing number may be also generalized to higher dimensions. Second, a variant of this crossing number is tightly connected to the well-known rectilinear crossing number (see Proposition 2.5 below). Third, one can find real problems which the Orchard crossing number can represent. For example, design a network of computers which should be constructed in a manner which allows possible extensions of the network in the future. Since we want to avoid (even future) crossings of the cables which are connecting between the computers, we need to count not only the present crossings, but also the separators (which might come to cross in the future).