Biclique immersions in graphs with independence number 2

Abstract

The analogue of Hadwiger's Conjecture for the immersion relation states that every graph~$G$ contains an immersion of $K_{\chi(G)}$. For graphs with independence number~2, this is equivalent to stating that every such $n$-vertex graph contains an immersion of $K_{\lceil n/2 \rceil}$. We show that every $n$-vertex graph with independence number~2 contains every complete bipartite graph on $\lceil n/2 \rceil$ vertices as an immersion.