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 $4v\le e\le 5v$ )."/> Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs | SpringerLink
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Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs

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 \(4v\le e\le 5v\)).

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Correspondence to Janos Pach, Rados Radoicic, Gabor Tardos or Geza Toth.

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Pach, J., Radoicic, R., Tardos, G. et al. Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs. Discrete Comput Geom 36, 527–552 (2006). https://doi.org/10.1007/s00454-006-1264-9

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  • DOI: https://doi.org/10.1007/s00454-006-1264-9

Keywords

  • Additive Constant
  • Parallel Edge
  • Triangular Face
  • Simple Face
  • Strong Neighbor