Determining the minimum size of maximal 1-plane graphs

TL;DR

This work determines the exact minimum edge count of maximal $1$-plane graphs of order $n$, proving $m(n)=\left\lceil\frac{7}{3}n\right\rceil-3$ for all $n\ge5$. It introduces the nest concept and a refined $K_4$-extension sequence with SWM$^*$-rules to control growth, yielding a tight global bound via a per-step inequality that aggregates to the $7/3$-density. The authors also construct a family of graphs $H_n$ achieving this bound, ensuring sharpness. The approach combines structural analysis of nests, hermits, and skeletons with detailed case analysis of $K_4$-links (strong, weak, micro) to secure the bound and its tightness. The results advance the understanding of edge-density in maximal $1$-planar graphs and resolve the asymptotically minimal size problem exactly for the stated range of $n$.

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 Barát and Tóth 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 $n\ge 5$.

Figures (20)

• Figure 1: A hermit $h$ surrounded by two pairs of crossing edges
• Figure 2: The nest ${\cal N}$ (the white region) at (a) is finite, and the nest (the white region) ${\cal N}$ at (b) is infinite, where the possible drawn edge $uv$ is omitted
• Figure 3: $F=G[\{x,y,z,u\}]\cong K_4$ and $|V(F)\cap V(G')|=3, 2$ or $1$, respectively
• Figure 4: $F_{i+1}$ is a strong-$K_4$-link from $G_i$ and $x\in V(F_{i+1})\setminus V(G_i)$
• Figure 5: Two types of nests in $\mathbb{W}(G_{i+1})\setminus \mathbb{W}(G_i)$, where a nest ${\cal N}$ in type B* is a special one in type B whose boundary $\hbox{bd}({\cal N})$ has two far half-edges incident with $x$

Theorems & Definitions (92)

• Theorem 1.1: Br
• Theorem 1.2: Br
• Theorem 1.3: BT
• Theorem 1.4
• Lemma 2.1: BTBr
• Lemma 2.2: BTBr
• Lemma 2.3: EH
• Lemma 2.4
• proof
• Lemma 2.5: BTBr