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 (graphs with rotation systems).
- 1.Archdeacon, D.: Problems in topological graph theory, http://www.emba.uvm.edu/~archdeac/problems/altcross.html(accessed April 7, 2005)
- 4.Chojnacki, C. (Haim Hanani): Uber wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundamenta Mathematicae 23, 135–142 (1934)Google Scholar
- 8.Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings (April 2005) (manuscript)Google Scholar
- 11.Pavel Valtr. On the pair-crossing number (manuscript)Google Scholar
- 12.West, D.: Open problems - graph theory and combinatorics., http://www.math.uiuc.edu/~west/openp/ (accessed April 7, 2005)