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 (Formula presented). 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. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1190, pp. 345–354). Springer Verlag. https://doi.org/10.1007/3-540-62495-3_59
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