We show that if a graph of v vertices can be drawn in the plane so that every edge crosses at most k > 0 others, then its number of edges cannot exceed 4.108√ci;kv. For k ≤ 4, we establish a better bound, (k + 3)(v - 2), which is tight for k = 1 and 2. We apply these estimates to improve a result of Ajtai et al. and Leighton, providing a general lower bound for the crossing number of a graph in terms of its number of vertices and edges.
CITATION STYLE
Pach, J., & Tóth, G. (1997). Graphs drawn with few crossings per edge. Combinatorica, 17(3), 427–439. https://doi.org/10.1007/BF01215922
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