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 \(n-2-(2\lambda _1+2\lambda _2+\lambda _3)/6\), where \(\lambda _1\) and \(\lambda _2\) are respectively the numbers of 2-degree and 4-degree vertices in G, and \(\lambda _3\) is the number of odd vertices w in G such that either \(d_G(w)\le 9\) 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 \(m-3n+6\).
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Acknowledgements
The authors would like to thank the anonymous referees for many helpful comments and suggestions.
Funding
The work was supported by the National Natural Science Foundation of China (no. 11301169), Hunan Provincial Natural Science Foundation of China (no. 2017JJ2055) and Scientific Research Fund of Hunan Provincial Education Department of China (no. 18A432).
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Ouyang, Z., Huang, Y. & Dong, F. The Maximal 1-Planarity and Crossing Numbers of Graphs. Graphs and Combinatorics 37, 1333–1344 (2021). https://doi.org/10.1007/s00373-021-02320-x
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DOI: https://doi.org/10.1007/s00373-021-02320-x