k-Quasi-Planar Graphs

A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A topological graph is simple if every pair of its edges intersect at most once (either at a vertex or at their intersection). In 1996, Pach, Shahrokhi. and Szegedy [16] showed that every n-vertex simple k-quasi-planar graph contains at most O(n(log n)(2k-4)) edges. This upper bound was recently improved (for large k) by Fox and Pach [8] to n(log n)(O(log k)). In this note, we show that all such graphs contain at most (n log(2) n)2(alpha ck(n)) edges, where alpha(n) denotes the inverse Ackermann function and c(k) is a constant that depends only on k.