The Number of Edges in k-Quasi-planar Graphs
Name
Fox-2013-The number of edges in k-quasi-planar graphs.pdf
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281.05 KB
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Author(s)
Fox, Jacob
Suk, Andrew
Pach, Janos
Alternative Title
The Number of Edges in $k$-Quasi-planar Graphs
Date Issued
March 2013
Journal
SIAM Journal on Discrete Mathematics
Publisher
Society for Industrial and Applied Mathematics
Citation
Fox, Jacob, János Pach, and Andrew Suk. “The Number of Edges in $k$-Quasi-planar Graphs.” SIAM Journal on Discrete Mathematics 27, no. 1 (March 19, 2013): 550-561. © 2013, Society for Industrial and Applied Mathematics
Version
Final published version
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)[superscript O(log k)]. In the present paper, we improve this bound to (n log n )2[superscript alpha(n) ck] 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[superscript ck 6]n log n.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/110858586
