Planar Crossing Numbers of Graphs of Bounded Genus

Abstract

Pach and Tóth proved that any n-vertex graph of genus g and maximum degree d has a planar crossing number at most c g dn, for a constant c > 1. We improve on this result by decreasing the bound to O(dgn), and also prove that our result is tight within a constant factor. Our proof is constructive and yields an algorithm with time complexity O(dgn). As a consequence of our main result, we show a relation between the planar crossing number and the surface crossing number.A drawing of a graph G in the plane is an injection of the set of the vertices of G onto points of the plane and a mapping of the set of the edges of G onto simple continuous curves such that the endpoints of each edge are mapped onto the endpoints of its image curve. Moreover, no curve should contain an image of a vertex in its inside and no three curves should intersect in the same point, unless it is an endpoint. The planar crossing number (or simply the crossing number) of G, denoted by cr(G), is the minimum number of edge crossings over all drawings of G in the plane.The concept of crossing numbers was introduced [26] more than 50 years ago by Turán. Although there have been scores of results and publications since, because of A preliminary version of this work was presented at ICALP'2006.H.N. Djidjev ( ) Los

The Graph Crossing Number and its Variants: A Survey

“…For Σ ∈ {S g , N g } we have cr Σ (G) = Ω(m 3 /n 2 ) if 0 g < n 2 /m and cr Σ (G) = Ω(m 2 /g) if n 2 /m g m/64 [492]. Asymptotically, cr(G) = O(g(cr Sg (G) + n)) for graphs of bounded degree as long as g = o(n) [176]. If cr Σ (G) = 0, then cr(G) c Σ ∆n, where ∆ is the maximum degree of G [97], for an algorithmic view of this result, see [135].…”

Computing the Stretch of an Embedded Graph

“…Unfortunately, Cabello and Mohar [3] show that the crossing number of such graphs is not computable in polynomial time, even when Σ is the torus. Djidjev and Vrt'o [4] show that the crossing number of G is upper bounded by O(g∆n), where n is the number of vertices in G and ∆ is the maximum degree of G. This is an improvement over the previous bound of O(C g ∆n), for some constant C, by Börözky, Pach and Tóth [1]. Under some mild assumptions about the density of the embedding of G, Hliněný and Chimani [5] give a (3 • 2 3g+2 ∆ 2 )approximation algorithm for the crossing number of G. This last work is the main motivation for our research.…”