Graphs Drawn with Few Crossings Per Edge

Given a simple graph G, let v(G) and e(G) denote its number of vertices and edges, respectively. We say that G is drawn in the plane if its vertices are represented by distinct points of the plane and its edges are represented by Jordan arcs connecting the corresponding point pairs but not passing through any other vertex. Throughout this paper, we only consider drawings with the property that any two arcs have at most one point in common. This is either a common endpoint or a common interior point where the two arcs properly cross each other. We will not make any notational distinction between vertices of G and the corresponding points in the plane, or between edges of G and the corresponding Jordan arcs. We address the following question. What is the maximum number of edges that a simple graph of v vertices can have if it can be drawn in the plane so that every edge crosses at most k others? For k = 0, i.e. for planar graphs, the answer is 3v ? 6. Our rst theorem generalizes this result to k 4. The case k = 1 has been discovered independently by Bernd GG artner, Torsten Thiele, and G unter Ziegler (personal communication). Theorem 1. Let G be a simple graph drawn in the plane so that every edge is crossed by at most k others. If 0 k 4, then we have e(G) (k + 3)(v(G) ? 2): For k = 0; 1; 2, the above bound cannot be improved. The crossing number cr(G) of a graph G is the minimum number of crossing pairs of edges, over all drawings of G in the plane. Ajtai et al. AC82] and, independently, Leighton L83] obtained a general lower bound for the crossing number of a graph, which found many applications in combinatorial geometry and in VLSI design (see PA95], PS96], S95]). Our next result, whose proof is based on Theorem 1, improves the bound of Ajtai et al. by roughly a factor of 2.