# Graphs drawn with few crossings per edge

@article{Pach1997GraphsDW, title={Graphs drawn with few crossings per edge}, author={J{\'a}nos Pach and G{\'e}za T{\'o}th}, journal={Combinatorica}, year={1997}, volume={17}, pages={427-439} }

We show that if a graph ofv vertices can be drawn in the plane so that every edge crosses at mostk>0 others, then its number of edges cannot exceed 4.108√kv. Fork≤4, we establish a better bound, (k+3)(v−2), which is tight fork=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.

## 251 Citations

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- MathematicsDiscret. Comput. Geom.
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It is shown that the number of crossing points, counted without multiplicity, is at least constant times e and that the order of magnitude of this bound cannot be improved.

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A tight bound of 4n − 9 on the maximum number of edges of such a graph for a straight-edge drawing is proved and generalizations to monotone graph properties are discussed.

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A conjecture of Erdos os and Guy is proved by showing that κ(n,e)n 2 /e 3 tends to a positive constant as n→∈fty and n l e l n 2 .

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It is shown that, for any k, if G has at least Ckn edges and n vertices, then it contains three sets of k edges, such that every edge in any of the sets crosses all edges in the other two sets.

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It is proved that many optimization problems, including maximum independent set, minimum vertex cover, minimum dominating set and many others, admit polynomial time approximation schemes when restricted to graphs such that each edge has a bounded number of crossings.

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Let G be a graph that can be drawn on a plane in such a way that any edge intersects at most one other edge. It is proved that the chromatic number of G does not exceed 7. The bound $ \chi (G)\leq…

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Let G be a bipartite graph without loops and multiple edges on v ≥ 4 vertices, which can be drawn on a plane in such a way that any edge intersects at most one other edge. It is proved that such a…

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