On the Decay of Crossing Numbers

The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [RT93] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that cr(G - e) k - c k. They showed only that G always has an edge e with cr(G - e) 2 5 cr(G) - O(1). We prove that for every fixed > 0, there is a constant n0 depending on such that if G is a graph with n > n0 vertices and m > n1+ edges, then G has a subgraph G with at most (1 - 1 24 )m edges such that cr(G ) ( 1 28 - o(1))cr(G).

• Type: Conference
• Year: 2006
• Where: GD
• Authors: Jacob Fox, Csaba D. Tóth
• Keywords: Computer Graphics, Drawings, GD 2006, Minimum Number