Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs |
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Authors: | Janos Pach Rados Radoicic Gabor Tardos Geza Toth |
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Institution: | (1) Courant Institute, N.Y.U., 251 Mercer Street, New York, NY 10012, USA;(2) Renyi Institute of Mathematics, Hungarian Academy of Sciences, Pf. 127, H-1364, Budapest, Hungary;(3) Department of Mathematics, Baruch College, CUNY, One Bernard Baruch Way, New York, NY 10010, USA |
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Abstract: | Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices
and e > 4v edges is at least ce3/v2, where c > 0 is an absolute constant. This result, known as the "Crossing Lemma," has found many important applications in
discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of
the constant by showing that the result holds with c > 1024/31827 > 0.032. The proof has two new ingredients, interesting
in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others,
then its number of edges cannot exceed 5.5(v-2); and (2) the crossing number of any graph is at least
Both bounds are tight up to an additive constant (the latter one in the range
). |
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