Improved upper bounds on the crossing number

Abstract

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.

Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface

“…Regarding this aspect, our Theorem 2.1 is much stronger. It is also worth to note that our theorem works for multigraphs, unlike, for instance, the estimate of [10].…”

“…Such results have, moreover, been strongly generalized towards all proper minor-closed graph families in [28,10]. The common limitation of all these four mentioned papers is that they give only upper bounds on the crossing number, which may be asymptotically optimal in the worst cases but, at the same time, are way above the real crossing number in other cases.…”

On crossing numbers of geometric proximity graphs

“…These problems have been attracting a great amount of attention and recently a continuous chain of improvements has led progressively to narrow the gap between the lower and upper bounds [15,3,2]. There are also several results on the numbers of crossings that are sensitive to the size of the graph -particulary the crossing lemma [4,17,6]-, or to the exclusion of some configurations [6,20,22,28,8].…”

A Golden Ratio Inequality for Vertex Degrees of Graphs

“…In a study of the crossing number of graphs [1,2], the authors proved upper bounds on the crossing number for various graph classes. For a graph G with vertex set V (G) and edge set E(G), if d(v) denotes the degree of each vertex v ∈ V (G), then these upper bounds are of the form…”

“…The proof is an easy exercise. While the exponent of 3 in the right-hand side of (1) cannot be improved for regular graphs, for classes of graphs that allow for many different vertex degrees, such as trees and planar graphs, it is natural to ask what is the minimum exponent such that every graph in the class satisfies an analogous inequality (allowing 1 2 to be replaced by some other constant).…”