The number of edges in k-quasi-planar graphs

TL;DR

This paper improves bounds on the maximum number of edges in k-quasi-planar graphs, especially for graphs with certain drawing constraints, advancing understanding of their combinatorial properties.

Contribution

It provides tighter upper bounds for the number of edges in k-quasi-planar graphs under specific drawing conditions, extending previous results.

Findings

Improved upper bound to (n log n)^{2^{α^{c_k}(n)}} for graphs with at most one crossing per pair of edges.

Established that k-quasi-planar graphs with x-monotone edges have at most 2^{ck^6} n log n edges.

Progressed towards the conjecture that such graphs have O(n) edges for fixed k.

Abstract

A graph drawn in the plane is called k-quasi-planar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is O(n). The best known upper bound is n(\log n)^{O(\log k)}. In the present note, we improve this bound to (n\log n)2^{\alpha^{c_k}(n)} in the special case where the graph is drawn in such a way that every pair of edges meet at most once. Here \alpha(n) denotes the (extremely slowly growing) inverse of the Ackermann function. We also make further progress on the conjecture for k-quasi-planar graphs in which every edge is drawn as an x-monotone curve. Extending some ideas of Valtr, we prove that the maximum number of edges of such graphs is at most 2^{ck^6}n\log n.