Improved upper bounds on the crossing number (original) (raw)

2008, Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08

The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph G that does not contain a fixed graph as a minor has crossing number O(∆n), where G has n vertices and maximum degree ∆. This dependence on n and ∆ is best possible. This result answers an open question of Wood and Telle [New York J. Mathematics, 2007], who proved the best previous bound of O(∆ 2 n). In addition, we prove that every K5-minor-free graph G has crossing number at most 2 P v deg(v) 2 , which again is the best possible dependence on the degrees of G. We also study the convex and rectilinear crossing numbers, and prove an O(∆n) bound for the convex crossing number of bounded pathwidth graphs, and a P v deg(v) 2 bound for the rectilinear crossing number of K3,3-minor-free graphs.

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