A simple proof of the existence of complete bipartite graph immersion in graphs with independence number two

TL;DR

This paper provides a simplified proof that graphs with independence number two contain immersions of all complete bipartite graphs on half their vertices, supporting a special case of Hadwiger's conjecture.

Contribution

It offers a much simpler proof of the known result that graphs with independence number two contain all complete bipartite graph immersions on half their vertices.

Findings

Graphs with independence number two contain all complete bipartite graph immersions on half their vertices.

The proof simplifies previous complex arguments.

Supports a special case of Hadwiger's conjecture.

Abstract

Hadwiger's conjecture for the immersion relation posits that every graph $G$ contains an immersion of the complete graph $K_{\chi(G)}$. Vergara showed that this is equivalent to saying that every $n$-vertex graph $G$ with $\alpha(G)=2$ contains an immersion of the complete graph on $\lceil\frac{n}{2}\rceil$ vertices. Recently, Botler et al. showed that every $n$-vertex graph $G$ with $\alpha(G)=2$ contains every complete bipartite graph on $\lceil\frac{n}{2}\rceil$ vertices as an immersion. In this paper, we give a much simpler proof of this result.