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- W2000052110 abstract "The crossing number cr( G ) of a graph G is the minimum possible number of edge crossings in a drawing of G in the plane, while the pair-crossing number pcr( G ) is the smallest number of pairs of edges that cross in a drawing of G in the plane. While cr( G )⩾pcr( G ) holds trivially, it is not known whether a strict inequality can ever occur (this question was raised by Mohar and Pach and Tóth). We aim at bounding cr( G ) in terms of pcr( G ). Using the methods of Leighton and Rao, Bhatt and Leighton, and Even, Guha and Schieber, we prove that cr (G)=O( log 3 n( pcr (G)+ ssqd (G))) , where n =| V ( G )| and ssqd (G)=∑ v∈V(G) deg G (v) 2 . One of the main steps is an analogy of the well-known lower bound cr (G)=Ω(b(G) 2 )−O( ssqd (G)) , where b ( G ) is the bisection width of G , that is, the smallest number of edges that have to be removed so that no component of the resulting graph has more than 2 3 n vertices. We show that pcr (G)=Ω(b(G) 2 / log 2 n)−O( ssqd (G)) . We also prove by similar methods that a graph G with crossing number k= cr (G)>C ssqd (G) m log 2 n has a nonplanar subgraph on at most O Δ nm log 2 n k vertices, where m is the number of edges, Δ is the maximum degree in G , and C is a suitable sufficiently large constant." @default.
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- W2000052110 date "2004-09-01" @default.
- W2000052110 modified "2023-10-16" @default.
- W2000052110 title "Crossing number, pair-crossing number, and expansion" @default.
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- W2000052110 doi "https://doi.org/10.1016/j.jctb.2003.09.002" @default.
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