Monotone Crossing Number

The monotone crossing number of G is defined as the smallest number of crossing points in a drawing of G in the plane, where every edge is represented by an x-monotone curve, that is, by a connected continuous arc with the property that every vertical line intersects it in at most one point. It is shown that this parameter can be strictly larger than the classical crossing number cr(G), but it is bounded from above by 2cr2(G). This is in sharp contrast with the behavior of the rectilinear crossing number, which cannot be bounded from above by any function of cr(G).

[1]  Michael J. Pelsmajer,et al.  Hanani–Tutte, Monotone Drawings, and Level-Planarity , 2013 .

[2]  P. Erdös,et al.  Crossing Number Problems , 1973 .

[3]  George E. Nicholson,et al.  Combinatorial Mathematics and Its Applications Year. , 1971 .

[4]  János Pach,et al.  Combinatorial and Computational Geometry , 2011 .

[5]  George Pólya.  A note of welcome , 1977, J. Graph Theory.

[6]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[7]  N. Sloane,et al.  Proof Techniques in Graph Theory , 1970 .

[8]  János Pach,et al.  Conway's Conjecture for Monotone Thrackles , 2011, Am. Math. Mon..

[9]  János Pach,et al.  Monotone drawings of planar graphs , 2002, J. Graph Theory.

[10]  Géza Tóth,et al.  Note on the Pair-crossing Number and the Odd-crossing Number , 2008, CCCG.

[11]  Michael J. Pelsmajer,et al.  Removing Independently Even Crossings , 2009, SIAM J. Discret. Math..

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

[13]  Michael J. Pelsmajer,et al.  Odd Crossing Number and Crossing Number Are Not the Same , 2008, Discret. Comput. Geom..

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

[15]  Frank Harary,et al.  Graph Theory , 2016 .

[16]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[17]  G. Tóth.  A Better Bound for the Pair-Crossing Number , 2013 .

[18]  Jj Anos Pach.  Which Crossing Number Is It Anyway? , 1998 .

[19]  P. Erdös,et al.  Crossing Number Problems , 1973 .

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

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

[22]  Pavel Valtr.  On the Pair-Crossing Number , 2007 .