Journal Articles

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.