Very few results are known which yield the crossing number of an infinite class of graphs on some surface. In this paper it is shown that by taking the class of graphs to be ¿-dimensional cubes Q(d) and by allowing the genus of the surface to vary, we obtain upper and lower bounds on the crossing numbers which are independent of d. Specifically, if the genus of the surface is always y(Q(d))—k, ...

A sufficient condition is given that a certain drawing minimizes the crossing number. The condition is in terms of intersections in an arbitrary set system related to the drawing, and is like a correlation inequality.

In the last years, several integer linear programming (ILP) formulations for the crossing number problem arose. While they all contain a common conceptual core, the properties of the corresponding polytopes have never been investigated. In this paper, we formally establish the crossing number polytope and show several facet-defining constraint classes.

It is proved that the crossing number of C6 X Cn is 4n for every n 2: 6. This is in agreement with the general conjecture that the crossing number of Cm x en is (m 2)n, for 3 ::; m :s; n.

R. Bruce Richter Department of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6 and Gelasio Salazar1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30319 and IICO{UASLP, San Luis Potosi, Mexico 78000 21 April 1999 Abstract. It is proved that the crossing number of the Generalized Petersen Graph P (3k+ h; 3) is k + h if h 2 f0; 2g and k + 3 if h = 1, f...

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c3(K), which is the minimum number of triple crossings in a projection of K. We o...

We show that computing the crossing number of a graph with a given rotation system is NP-complete. This result leads to a new and much simpler proof of Hliněný’s result, that computing the crossing number of a cubic graph (no rotation system) is NP-complete.

We show that, if P6=NP, there is a constant c0 > 1 such that there is no c0approximation algorithm for the crossing number, even when restricted to 3-regular graphs.

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for smallgenus surfaces. We prove that all of the commonly con...

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the Euclidean plane. The k-planar crossing number crk(G) of G is min{cr(G1) + cr(G2) + . . .+ cr(Gk)}, where the minimum is taken over all possible decompositions of G into k subgraphs G1, G2, . . . , Gk. The problem of computing the crossing number of complete graphs, cr(Kn), exactly for sm...