The minimum crossing number and minimum size of maximal 1-plane graphs with given connectivity

TL;DR

This work addresses the extremal problem for $k$-connected maximal $1$-plane graphs with $n$ vertices by deriving sharp lower bounds on the crossing number $cr(G)$ and on the edge count $|E(G)|$ for each $k$ in $\{3,4,5,6,7\}$. The authors develop a framework based on triangulations of the crossing-augmented graph $G^\times$ and face-graph analysis to prove piecewise bounds, and they construct explicit extremal graphs (e.g., $XH^k$, $YH^k$, $XM^k$, and the $H^k$-based families) that realize the bounds, establishing tightness in many cases. For $k=7$, they obtain a strong bound $cr(G)\ge \tfrac{3}{4}n$, demonstrated via crossing-incidence counting, with concrete 7-connected examples at $n=24$ and $n=56$. The results extend prior bounds for maximal $1$-plane graphs and raise open questions about the extremal behavior for lower connectivity regimes and broader classes of maximal $1$-plane graphs.

Abstract

A 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no additional edge can be added without violating 1-planarity or simplicity. It is known that any maximal 1-plane graph is $k$-connected for some $k$ with $2\le k\le 7$. Recently, Huang et al. proved that any maximal 1-plane graph with $n$ ($\ge 5$) vertices has at least $\lceil\frac{7}{3}n\rceil-3$ edges, which is tight for all integers $n\ge 5$. In this paper, we study $k$-connected maximal 1-plane graphs for each $k$ with $3\le k\le 7$, and establish a lower bound for their crossing numbers and a lower bound for their edge numbers, respectively.

Figures (8)

• Figure 1: A face $F$ of $G^\times$ bounded by a facial cycle $C$ with at least four vertices
• Figure 2: Auxiliary graphs for proving Lemma \ref{['trface']}
• Figure 3: Three triangulation operations
• Figure 4: Plane graphs $H^3$ and $H^3 \circ H^3$, where the structure outside the green circle $C_{2^{4}}$ is omitted
• Figure 5: The 1-plane graphs $XH^1$ and $YH^1$

Theorems & Definitions (43)

• Theorem 1: FJ
• Theorem 2: JB
• Theorem 3: HOZD
• Theorem 4
• Theorem 5
• Lemma 1: JBFJ
• Lemma 2: JBFJ
• Lemma 3: OHD
• Lemma 4
• proof