Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs

  • Authors:

  • Janos Pach;Rados Radoicic;Gabor Tardos;Geza Toth

  • Affiliations:

  • Courant Institute, N.Y.U., 251 Mercer Street, New York, NY 10012, USA and Renyi Institute of Mathematics, Hungarian Academy of Sciences, Pf. 127, H-1364, Budapest, Hungary;Department of Mathematics, Baruch College, CUNY, One Bernard Baruch Way, New York, NY 10010, USA;Renyi Institute of Mathematics, Hungarian Academy of Sciences, Pf. 127, H-1364, Budapest, Hungary;Renyi Institute of Mathematics, Hungarian Academy of Sciences, Pf. 127, H-1364, Budapest, Hungary

  • Venue:
  • Discrete & Computational Geometry
  • Year:
  • 2006

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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 $\frac73e-\frac{25}3(v-2).$ Both bounds are tight up to an additive constant (the latter one in the range $4v\le e\le 5v$).