Improving the Crossing Lemma by finding more crossings in sparse graphs
نویسندگان
چکیده
Twenty years ago, Ajtai, Chvátal, Newborn, Szemerédi, and, independently, Leighton discovered that the crossing number of any graph with v vertices and e > 4v edges is at least ce/v, 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 on their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most 3 others, then its number of edges cannot exceed 5.5(v−2); and (2) the crossing number of any graph is at least 7 3 e− 25 3 (v−2). Both bounds are tight up to an additive constant (the latter one in the range 4v ≤ e ≤ 5v).
منابع مشابه
On topological graphs with at most four crossings per edge
We show that if a graph G with n ≥ 3 vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then G has at most 6n− 12 edges. This settles a conjecture of Pach, Radoičić, Tardos, and Tóth. As a corollary we also obtain a better bound for the Crossing Lemma which gives a lower bound for the minimum number of crossings in a drawing of a graph.
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