The maximum number of edges in an outer k-planar graph on n vertices is at most (sqrt(2)+epsilon) sqrt(k) n plus a lower-order term, with epsilon tending to 0 as k grows.
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On the maximum number of edges of outer k-planar graphs
The maximum number of edges in an outer k-planar graph on n vertices is at most (sqrt(2)+epsilon) sqrt(k) n plus a lower-order term, with epsilon tending to 0 as k grows.