Crossing numbers of imbalanced graphs

The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. According to the Crossing Lemma of M. Ajtai, V. Chvátal, M. Newborn, E. Szemerédi, Theory and Practice of Combinatorics, North‐Holland, Amsterdam, New York, 1982, pp. 9–12 and F. T. Leighton, Complexity Issues in VLSI, MIT Press, Cambridge, 1983, the crossing number of any graph with n vertices and e>4n edges is at least constant times e³/n². Apart from the value of the constant, this bound cannot be improved. We establish some stronger lower bounds under the assumption that the distribution of the degrees of the vertices is irregular. In particular, we show that if the degrees of the vertices are d₁?d₂?···?d_n, then the crossing number satisfies begin{eqnarray*}{rm{cr}}(G)geq frac{c_{1}}{n}end{eqnarray*} with begin{eqnarray*}{textstylesumnolimits_{{{i}}={{{1}}}}^{{{n}}}}{{id}}_{{{i}}}^{{{3}}}-{{c}}_{{{2}}}{{n}}^{{{2}}}end{eqnarray*}, and that this bound is tight apart from the values of the constants c₁, c₂>0. Some applications are also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 12–21, 2010