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...

We revoke the problem of drawing graphs in the plane so that only certain specified pairs of edges are allowed to cross. We overview some previous results and open problems, namely the connection to intersection graphs of curves in the plane. We complement these by stating a new conjecture and showing that its proof would solve the problem of algorithmic decidability of recognition of string gr...

In this paper, we construct a new topological quantum field theory of cohomological type and show that its partition function is a crossing number.

One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1♯K2) = c(K1) + c(K2) always hold, where c(K) denotes the crossing number of a knot K and K1♯K2 is the connected sum of two (oriented) knots K1 and K2? The inequality c(K1♯K2) ≤ c(K1) + c(K2) is trivial, but very little more is known in general...