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- W2072233942 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 paper, we improve this bound to $(nlog n )2^{alpha(n)^{c_k}}$ 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 $2^{ck^6}nlog n$." @default.
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- W2072233942 date "2013-01-01" @default.
- W2072233942 modified "2023-10-10" @default.
- W2072233942 title "The Number of Edges in $k$-Quasi-planar Graphs" @default.
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- W2072233942 doi "https://doi.org/10.1137/110858586" @default.
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