On the crossing numbers of Cartesian products with trees

Zip product was recently used in a note establishing the crossing number of the Cartesian product K1,n2Pm. In this paper, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to weaken the connectivity condition under which the crossing number is additive for the zip product. Next, we deduce a general theorem for bounding the crossing numbers of (capped) Cartesian products of graphs with trees, which yields exact results under certain symmetry conditions. We apply this theorem to obtain exact and approximate results on crossing numbers of various graphs with trees.

[1]  Michael J. Pelsmajer,et al.  Odd Crossing Number Is Not Crossing Number , 2005, GD.

[2]  R. Bruce Richter,et al.  The crossing number of c4 × c4 , 1995, J. Graph Theory.

[3]  Douglas R. Woodall,et al.  Cyclic-order graphs and Zarankiewicz's crossing-number conjecture , 1993, J. Graph Theory.

[4]  Marián Klešč.  On the crossing numbers of Cartesian products of stars and paths or cycles , 1991 .

[5]  R. Bruce Richter,et al.  Arrangements, circular arrangements and the crossing number of C7X Cn , 2004, J. Comb. Theory B.

[6]  R. Bruce Richter,et al.  The crossing number of C5 × Cn , 1996, J. Graph Theory.

[7]  Drago Bokal,et al.  On the crossing numbers of Cartesian products with paths , 2007, J. Comb. Theory, Ser. B.

[8]  Marián Klesc,et al.  The crossing numbers of products of paths and stars with 4-vertex graphs , 1994 .

[9]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[10]  Marián Klesc.  The crossing number of K2, 3xC3 , 2002, Discret. Math..

[11]  Lowell W. Beineke,et al.  On the crossing numbers of products of cycles and graphs of order four , 1980, J. Graph Theory.

[12]  Gelasio Salazar,et al.  The crossing number of Cm × Cn is as conjectured for n ≥ m(m + 1) , 2004, J. Graph Theory.

[13]  Gelasio Salazar,et al.  The crossing number of C6 × Cn , 2001, Australas. J Comb..

[14]  Marián Klesc,et al.  The crossing numbers of Cartesian products of paths with 5-vertex graphs , 2001, Discret. Math..

[15]  Daniel J. Kleitman,et al.  The crossing number of K5,n , 1970 .

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

[17]  R. Richter,et al.  The crossing number of C 5 × C n , 1996 .

[18]  Jesús Leaños,et al.  ON THE ADDITIVITY OF CROSSING NUMBERS OF GRAPHS , 2008 .

[19]  Carsten Thomassen,et al.  Intersections of curve systems and the crossing number ofC5×C5 , 1995, Discret. Comput. Geom..

[20]  Kouhei Asano,et al.  The crossing number of K1, 3, n and K2, 3, n , 1986, J. Graph Theory.

[21]  F. Shahrokhi,et al.  The Crossing Number of a Graph on a Compact 2-Manifold , 1996 .

[22]  G. Salazar,et al.  The crossing number of C m × C n is as conjectured for n ≥ m(m + 1) , 2004 .

[23]  R. Bruce Richter,et al.  Crossing numbers of sequences of graphs II: Planar tiles , 2003, J. Graph Theory.