The density of maximal IC-plane graphs and maximal NIC-plane graphs

TL;DR

This paper establishes tight bounds on the minimum number of edges in maximal IC-plane and NIC-plane graphs, providing new insights into their structural density for infinitely many graph sizes.

Contribution

It introduces tight lower bounds on edges for maximal IC-plane and NIC-plane graphs, advancing understanding of their density properties.

Findings

Maximal IC-plane graphs have at least (7/3)n - 14/3 edges.

Maximal NIC-plane graphs have at least (11/5)n - 18/5 edges.

Both bounds are tight for infinitely many values of n.

Abstract

In this paper, we show that any maximal IC-plane graph of order $n$ has at least $\left\lceil\frac{7}{3}n-\frac{14}{3}\right\rceil$ edges, and any maximal NIC-plane graph of order $n$ has at least $\left\lceil\frac{11}{5}n-\frac{18}{5}\right\rceil$ edges. Moreover, we show that both results are tight for infinitely many integers $n$.