On the Number of Edges in Maximally Linkless Graphs

TL;DR

This paper introduces a new family of maximally linkless graphs with fewer edges than previously known, specifically with at most (14/5)n edges, advancing understanding of their structural limits.

Contribution

The authors present a novel family of maximally linkless graphs with a tighter upper bound on the number of edges, improving upon the known maximum of 3n-3 edges.

Findings

New family of maximally linkless graphs with m ≤ (14/5)n edges

Improved upper bound on edges in maximally linkless graphs

Enhanced understanding of the structural properties of linkless graphs

Abstract

A maximally linkless graph is a graph that can be embedded in $\mathbb{R}^3$ without any links, but cannot be embedded in such a way if any other edge is added to the graph. Recently, a family of maximally linkless graphs was found with $m=3n-3$ edges. We improve upon this by demonstrating a new family of maximally linkless graphs with $m\le \frac{14}{5}n$ edges.