Complete bipartite immersion in graphs with independence number two: A simple proof

Abstract

A conjecture akin to Hadwiger's conjecture posits that every graph G contains an immersion of the complete graph $K_{\chi \left(G\right)}$. Vergara showed that, for every n-vertex graph G with independence number two, this is equivalent to saying that G 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 $\alpha \left(G\right)=2$ 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.