Progress on Crossing Number Problems

Crossing numbers have drawn much attention in the last couple of years and several surveys [22], [28], [33], problem collections [26], [27], and bibliographies [40] have been published. The present survey tries to give pointers to some of the most significant recent developments and identifies computational challenges.

[1]  J. Pach,et al.  Thirteen problems on crossing numbers , 2000 .

[2]  Robin Wilson,et al.  Graph theory and combinatorics , 1979 .

[3]  J. Pach.  Towards a Theory of Geometric Graphs , 2004 .

[4]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[5]  W. T. Tutte.  Toward a theory of crossing numbers , 1970 .

[6]  Sudipto Guha,et al.  Improved approximations of crossings in graph drawings , 2000, STOC '00.

[7]  Nathaniel Dean,et al.  Bounds for rectilinear crossing numbers , 1993, J. Graph Theory.

[8]  Paul Turán,et al.  A note of welcome , 1977, J. Graph Theory.

[9]  László A. Székely,et al.  Outerplanar Crossing Numbers, the Circular Arrangement Problem and Isoperimetric Functions , 2004, Electron. J. Comb..

[10]  E. Szemerédi,et al.  Crossing-Free Subgraphs , 1982 .

[11]  János Pach,et al.  Which Crossing Number Is It Anyway? , 1998, J. Comb. Theory, Ser. B.

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

[13]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

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

[15]  Sudipto Guha,et al.  Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas , 2002, SIAM J. Comput..

[16]  Martin Aigner,et al.  Proofs from THE BOOK , 1998 .

[17]  Martin Grohe,et al.  Computing crossing numbers in quadratic time , 2000, STOC '01.

[18]  Bernardo M. Ábrego,et al.  A Lower Bound for the Rectilinear Crossing Number , 2005, Graphs Comb..

[19]  József Balogh,et al.  k-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn , 2006, Discret. Comput. Geom..

[20]  E. Welzl,et al.  Convex Quadrilaterals and k-Sets , 2003 .

[21]  J. Pach,et al.  Combinatorial geometry , 1995, Wiley-Interscience series in discrete mathematics and optimization.

[22]  A. Owens,et al.  On the biplanar crossing number , 1971 .

[23]  János Pach,et al.  Combinatorial Geometry , 2012 .

[24]  János Pach,et al.  2 Two Important Bounds and Their Applications , 1994 .

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

[26]  László A. Székely,et al.  Crossing Numbers and Hard Erdős Problems in Discrete Geometry , 1997, Combinatorics, Probability and Computing.

[27]  E. Szemerédi,et al.  Unit distances in the Euclidean plane , 1984 .

[28]  László A. Székely.  A successful concept for measuring non-planarity of graphs: the crossing number , 2004, Discret. Math..

[29]  Franz Aurenhammer,et al.  On the Crossing Number of Complete Graphs , 2002, SCG '02.

[30]  György Elekes,et al.  On the number of sums and products , 1997 .

[31]  Endre Szemerédi,et al.  Extremal problems in discrete geometry , 1983, Comb..

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

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