标题： 给定连通性的最小交叉数和最大1平面图的最小大小 显示英文标题

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

摘要： 一个1-平面图是在平面上绘制的图，使得每条边最多被交叉一次。 如果以这种方式绘制一个1-平面图，该绘制称为一个{\it 平面图}。 如果无法添加任何额外的边而不破坏1-平面性或简单性，则该图是极大1-平面（或1-平面）的。 已知任何极大1-平面图对于某个$k$以及$2\le k\le 7$是$k$-连通的。 最近，黄等人。 证明了任何具有$n$($\ge 5$) 个顶点的最大1平面图至少有$\lceil\frac{7}{3}n\rceil-3$条边，这对于所有整数$n\ge 5$都是紧的。在本文中，我们研究了每个$k$具有$3\le k\le 7$的$k$-连通的最大1平面图，并分别建立了它们的交叉数的下界和边数的下界。 显示英文摘要

摘要： 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.