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 Ajtai, Chvátal, Newborn, Szemerédi [ACNS82] and Leighton [L83], the crossing number of any graph with n vertices and e > 4n edges is at least constant times e/n. 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 ≥ . . . ≥ dn, then the crossing number satisfies cr(G) ≥ c1 n ∑n i=1 idi − c2n, and that this bound is tight apart from the values of the constants c1, c2 > 0. Some applications are also presented.

[1]  Csaba D. Tóth,et al.  Distinct Distances in the Plane , 2001, Discret. Comput. Geom..

[2]  L. A. Oa,et al.  Crossing Numbers and Hard Erd} os Problems in Discrete Geometry , 1997 .

[3]  Gabor Tardos,et al.  A new entropy inequality for the Erd os distance problem , 2004 .

[4]  Csaba D. Tóth,et al.  On the Decay of Crossing Numbers , 2006, Graph Drawing.

[5]  F. Thomas Leighton,et al.  Complexity Issues in VLSI , 1983 .

[6]  Csaba D. Tóth,et al.  The k Most Frequent Distances in the Plane , 2002, Discret. Comput. Geom..

[7]  L. A S Z L,et al.  Crossing Numbers and Hard Erdős Problems in Discrete Geometry , 1997 .

[8]  Boris Aronov,et al.  Results on k-sets and j-facets via continuous motion , 1998, SCG '98.

[9]  J. Spencer,et al.  New Bounds on Crossing Numbers , 2000 .

[10]  János Pach,et al.  Graphs drawn with few crossings per edge , 1997, Comb..

[11]  Micha Sharir,et al.  k-Sets in Four Dimensions , 2006, Discret. Comput. Geom..

[12]  Artur Andrzejak.  Halving Point Sets , 1998 .

[13]  Tamal K. Dey,et al.  Improved Bounds for Planar k -Sets and Related Problems , 1998, Discret. Comput. Geom..

[14]  David S. Johnson,et al.  Crossing Number is NP-Complete , 1983 .

[15]  László A. Székely,et al.  Graph minors and the crossing number of graphs , 2007, Electron. Notes Discret. Math..

[16]  János Pach,et al.  Isosceles Triangles Determined by a Planar Point Set , 2002, Graphs Comb..

[17]  Csaba D. Tóth,et al.  A bipartite strengthening of the Crossing Lemma , 2007, J. Comb. Theory, Ser. B.

[18]  Micha Sharir,et al.  On the Number of Incidences Between Points and Curves , 1998, Combinatorics, Probability and Computing.

[19]  János Pach,et al.  Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs , 2006, Discret. Comput. Geom..