Improving the crossing lemma by finding more crossings in sparse graphs: [extended abstract]

Twenty years ago, Ajtai, Chvátal, Newborn, Szemerédi, and, independently, Leighton discovered that the crossing number of any graph with <i>v</i> vertices and <i>e</i>>4<i>v</i> edgesis at least <i>ce</i>³/<i>v</i>², where <i>c</i>>0 is an absolute constant. This result, known as the 'Crossing Lemma,' has found many important applications in discrete and computational geometry. It is tightup to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with <i>c</i>>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(<i>v</i>-2); and (2) the crossing number of any graph is at least <box>7<over>3</box>e - <box>25<over>3</box>(<i>v</i>-2). Both bounds are tight up to anadditive constant (the latter one in the range 4<i>v</i> ≤ <i>e</i> ≤ 5<i>v</i>).