The extremal function for bipartite linklessly embeddable graphs

TL;DR

This paper establishes an upper bound on the number of edges in bipartite linklessly embeddable graphs, showing they have at most 3n-10 edges unless they are a specific complete bipartite graph.

Contribution

It provides the first extremal function for bipartite linklessly embeddable graphs, characterizing their maximum edge count.

Findings

Bipartite linklessly embeddable graphs have at most 3n-10 edges.

Equality holds only for the complete bipartite graph K_{3,n-3}.

The result extends understanding of graph embeddings in 3-space.

Abstract

An embedding of a graph in $3$-space is linkless if for every two disjoint cycles there exists an embedded ball that contains one of the cycles and is disjoint from the other. We prove that every bipartite linklessly embeddable (simple) graph on $n\ge5$ vertices has at most $3n-10$ edges, unless it is isomorphic to the complete bipartite graph $K_{3,n-3}$.