Odd Crossing Number Is Not Crossing Number

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 on the annulus. 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 T´oth [4] and Sz´ekely [5] discuss the historical development of various interpretations and, often implicit, definitions of the crossing number concept. A drawing D of a graph...

• Type: Conference
• Year: 2005
• Where: GD
• Authors: Michael J. Pelsmajer, Marcus Schaefer, Daniel Stefankovic
• Keywords: GD 2005, Minimum Number, Odd Crossing Number, Plane Drawing