Improved bounds for the crossing number of the mesh of trees graph, Mn, are derived. In particular, we derive a new lower bound of 5n log n−44n 80 ‡ which improves on the previous bound of Leighton [11] by a constant factor, and an upper bound of (log n− 10 3 )n 2 + 8n− 20 3 . In addition, we construct drawings of Mn which achieve the upper bound number of crossings. We also prove that the cros...

The best lower bound known on the crossing number of the complete bipartite graph is : cr(Km,n) ≥ (1/5)(m)(m − 1)bn/2cb(n − 1)/2c In this paper we prove that: cr(Km,n) ≥ (1/5)m(m − 1)bn/2cb(n − 1)/2c + 9.9 × 10−6m2n2 for sufficiently large m and n.

The bipartite crossing number of a bipartite graph is the minimum number of crossings of edges when the partitions are placed on two parallel lines and edges are drawn as straigh line segments between the lines. We prove exact results, asymtotics and new upper bounds for the bipartite crossing numbers of 2-dimensional mesh graphs. Especially we show that bcr(P6 × Pn) = 35n − 47, for n ≥ 7.

We characterize Pfaffian graphs in terms of their drawings in the plane. We generalize the techniques used in the proof of this characterization, and prove a theorem about the numbers of crossings in T -joins in different drawings of a fixed graph. As a corollary we give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K2j+1 and K2j+1,2k+1.

The minor crossing number of a graph G is the minimum crossing number of a graph that contains G as a minor. It is proved that for every graph H there is a constant c, such that every graph G with no H-minor has minor crossing number at most c|V (G)|.

The n crossing number of a graph G, denoted crn(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a > b > 0, there exists a graph G for which cr0(G) = a, cr1(G) = b, and cr2(G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.

We show that under suitable non-degeneracy conditions, m points and n 2–dimensional algebraic surfaces in R satisfying certain “pseudoflat” requirements can have at most O ( mn + m + n ) incidences, provided that m ≤ n2− for any > 0 (where the implicit constant in the above bound depends on ), or m ≥ n. As a special case, we obtain the Szemerédi-Trotter theorem for 2–planes in R, again provided...