A sufficient condition for intrinsic knotting of bipartite graphs

TL;DR

This paper investigates conditions under which bipartite graphs are intrinsically knotted, providing proofs for specific cases and establishing bounds involving the number of vertices and edges.

Contribution

It introduces a conjecture on intrinsic knotting in bipartite graphs and proves it for graphs with five or six vertices in one part, also establishing bounds with a constant C_n.

Findings

Proved the conjecture for graphs with exactly five or six vertices in one part.

Established a bound involving a constant C_n for larger graphs.

Classified bipartite graphs with ten or fewer vertices regarding intrinsic knotting.

Abstract

We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six vertices in one part. We also show that there is a constant C_n such that a bipartite graph with exactly n \geq 5 vertices in one part and |E(G)| \geq 4 |V(G)| + C_n is intrinsically knotted. Finally, we classify bipartite graphs with ten or fewer vertices with respect to intrinsic knotting.