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- W2513288550 abstract "A k-planar graph is a graph that can be drawn in the plane such that every edge is crossed at most k times. For (k le 4), Pach and Toth [20] proved a bound of ((k+3)(n-2)) on the total number of edges of a k-planar graph, which is tight for (k=1,2). For (k=3), the bound of (6n-12) has been improved to (frac{11}{2}n-11) in [19] and has been shown to be optimal up to an additive constant for simple graphs. In this paper, we prove that the bound of (frac{11}{2}n-11) edges also holds for non-simple 3-planar graphs that admit drawings in which non-homotopic parallel edges and self-loops are allowed. Based on this result, a characterization of optimal 3-planar graphs (that is, 3-planar graphs with n vertices and exactly (frac{11}{2}n-11) edges) might be possible, as to the best of our knowledge the densest known simple 3-planar is not known to be optimal." @default.
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- W2513288550 date "2016-01-01" @default.
- W2513288550 modified "2023-09-27" @default.
- W2513288550 title "On the Density of Non-simple 3-Planar Graphs" @default.
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- W2513288550 doi "https://doi.org/10.1007/978-3-319-50106-2_27" @default.
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