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arXiv · 2502.11696

Determining the minimum size of maximal 1-plane graphs

Yuanqiu Huang·Zhangdong Ouyang·Licheng Zhang·Fengming Dong
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Abstract

A 1-plane graph is a graph together with a drawing in the plane in such a way that each edge is crossed at most once. A 1-plane graph is maximal if no edge can be added without violating either 1-planarity or simplicity. Let $m(n)$ denote the minimum size of a maximal $1$-plane graph of order $n$. Brandenburg et al. established that $m(n)\ge 2.1n-\frac{10}{3}$ for all $n\ge 4$, which was improved by Bar\'{a}t and T\'{o}th to $m(n)\ge \frac{20}{9}n-\frac{10}{3}$. In this paper, we confirm that $m(n)=\left\lceil\frac{7}{3}n\right\rceil-3$ for all $n\ge 5$.

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Yuanqiu Huang, Zhangdong Ouyang, Licheng Zhang, Fengming Dong. 2025-02-17. Determining the minimum size of maximal 1-plane graphs. https://arxiv.org/abs/2502.11696

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