The crossing number of C3 × Cn

The crossing number of C6 × Cn

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.

A lower bound for the crossing number of Cm × Cn

The crossing number of Cm × Cn is as conjectured for n >= m(m + 1)

It has been long conjectured that the crossing number of Cm × Cn is (m−2)n, for allm,n such that n ≥ m ≥ 3. In this paper it is shown that if n ≥ m(m+1) andm ≥ 3, then this conjecture holds. That is, the crossing number of Cm × Cn is as conjectured for all but finitely many n, for each m. The proof is largely based on techniques from the theory of arrangements, introduced by Adamsson and furthe...

Intersection of Curves and Crossing Number of Cm x Cn on Surfaces

Let (K 1 , K 2) be two families of closed curves on a surface S, such that |K 1 | = m, |K 2 | = n, m 0 ≤ m ≤ n, each curve in K 1 intersects each curve in K 2 , and no point of S is covered three times. When S is the plane, the projective plane, or the Klein bottle, we prove that the total number of intersections in K 1 ∪ K 2 is at least 10mn/9, 12mn/11, and mn + 10 −13 m 2 , respectively. More...

Analogies between the crossing number and the tangle crossing number

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tangl...