On the Stable Crossing Number of Cubes

Very few results are known which yield the crossing number of an infinite class of graphs on some surface. In this paper it is shown that by taking the class of graphs to be ¿-dimensional cubes Q(d) and by allowing the genus of the surface to vary, we obtain upper and lower bounds on the crossing numbers which are independent of d. Specifically, if the genus of the surface is always y(Q(d))—k, where y(Q(d)) is the genus of Q(d) and k is a fixed nonnegative integer, then 4k^crytQ(d))_t.(Q(d))^Sk provided that k is not too large compared to d. 0. Introduction. In this paper, we shall deal only with finite graphs without loops or parallel edges. If G is a graph, we denote by V(G) its set of vertices and E(G) its set of edges. By linear algebra, we can realize any graph G as a subset of three-dimensional euclidean space—simply put the set V(G) of vertices in general position (no 3 collinear, no 4 coplanar) and put in the edges of £(G) as straight line segments. Obviously, any two such realizations, with the subspace topologies, are homeomorphic. By the surface S9 of genus g, g^O, we mean the closed orientable 2manifold obtained by attaching g handles to the sphere—e.g., 1ZX=torus. An immersion or -\L\, where K and L are finite simplicial complexes with geometric realizations \K\ and \L\, is simplicial when it carries simplexes linearly into simplexes and is PL when there are Received by the editors February 19, 1971 and, in revised form, March 1, 1972. AMS 1970 subject classifications. Primary 05C10; Secondary 55A15.