Dong, F. M.

A 1-planar graph is a graph which has a drawing on the plane such that each edge has at most one crossing. Czap and Hudák showed that every 1-planar graph with n vertices has crossing number at most n−2. In this paper, we prove that every maximal 1-planar graph G with n vertices has crossing number at most n−2−(2λ1+2λ2+λ3)/6, where λ1 and λ2 are respectively the numbers of 2-degree and 4-degree vertices in G, and λ3 is the number of odd vertices w in G such that either dG(w)≤9 or G−w is 2-connected. Furthermore, we show that every 3-connected maximal 1-planar graph with n vertices and m edges has crossing number m−3n+6.

A graph G is k-crossing-critical if cr(G) ≥ k, but cr(G \ e) < k for each edge e ∈ E(G), where cr(G) is the crossing number of G. It is known that the latest upper bound of cr(G) for a k-crossing-critical graph G is 2k+8 √ k+47 when δ(G) ≥ 3, and 2k+35 when δ(G) ≥ 4, where δ(G) is the minimum degree of G. In this paper, we mainly show that for any k-crossing-critical graph G with n vertices, cr(G) ≤ 2k+8 when δ(G) ≥ 4, and cr(G) ≤ 2k− √ k/2n+35/6 when δ(G) ≥ 5.

D. Bokal proved that the crossing number is additive for the zip product under the condition of having two coherent bundles in the zipped graphs. This property is very effective when dealing with the crossing numbers of (capped) Cartesian product of trees with graphs containing a dominating vertex. In this paper, we first prove that the crossing number is still additive for the zip product under a weaker condition. Based on the new condition, we then establish some general expressions for bounding the crossing numbers of (capped) Cartesian product of trees with graphs (possibly without dominating vertex). Exact values of the crossing numbers of Cartesian product of trees with most graphs of order at most five are obtained by applying these expressions, which extend some previous results due to M. Klesc. In fact, our results can also be applied to deal with Cartesian product of trees with graphs of order more than five.