Odd Crossing Number and Crossing Number Are Not the Same

The crossing number of a graph is the minimum number of edge intersections in a plane drawing of a graph, where each intersection is counted separately. If instead we count the number of pairs of edges that intersect an odd number of times, we obtain the odd crossing number. We show that there is a graph for which these two concepts differ, answering a well-known open question on crossing numbers. To derive the result we study drawings of maps (graphs with rotation systems). 1 A Confusion of Crossing Numbers Intuitively, the crossing number of a graph is the smallest number of edge crossings in any plane drawing of the graph. As it turns out, this definition leaves room for interpretation, depending on how we answer the questions: what is a drawing, what is a crossing, and how do we count crossings? The papers by Pach and Tóth [7] and Székely [9] discuss the historical development of various interpretations and definitions—often implicit—of the crossing number concept. A drawing D of a graph G is a mapping of the vertices and edges of G to the Euclidean plane, associating a distinct point with each vertex, and a simple plane curve with each edge so that the ends of an edge map to the endpoints of the corresponding curve. For simplicity, we also require that • a curve does not contain any endpoints of other curves in its interior, • two curves do not touch (that is, intersect without crossing), and • no more than two curves intersect in a point (other than at a shared endpoint).