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).
Unable to display preview. Download preview PDF.
- 1.Archdeacon, D.: Problems in topological graph theory. http://www.emba.uvm.edu/~archdeac/problems/altcross.html (accessed April 7th, 2005)
- 6.Pach, J.: Crossing numbers. In: Discrete and Computational Geometry (Tokyo, 1998). Lecture Notes in Comput. Sci., vol. 1763, pp. 267–273. Springer, Berlin (2000) Google Scholar
- 8.Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings. In: EuroComb, April 2005 Google Scholar
- 11.Valtr, P.: On the pair-crossing number. In: Combinatorial and Computational Geometry. MSRI Publications, vol. 52, pp. 545–551 (2005) Google Scholar
- 12.West, D.: Open problems—graph theory and combinatorics. http://www.math.uiuc.edu/~west/openp/ (accessed April 7th, 2005)