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The maximal 1-planarity and crossing numbers of graphs
Citation
Ouyang, Z., Huang, Y., & Dong, F. (2021). The maximal 1-planarity and crossing numbers of graphs. Graphs and Combinatorics, 37, 1333-1344. https://doi.org/10.1007/s00373-021-02320-x
Abstract
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.
Date Issued
2021
Publisher
Springer Nature
Journal
Graphs and Combinatorics
DOI
10.1007/s00373-021-02320-x
Grant ID
Grant no.: 11301169
Grant no.: 2017JJ2055
Grant no.: 18A432
Funding Agency
National Natural Science Foundation of China
Hunan Provincial Natural Science Foundation of China
Scientific Research Fund of Hunan Provincial Education Department of China