Large Immersions in Graphs with Independence Number 3 and 4

Abstract

The analogue of Hadwiger's conjecture for the immersion order states that every graph G contains K χ(G) as an immersion. If true, it would imply that every graph with n vertices and independence number α contains K ⌈ n α ⌉ as an immersion. The best currently known bound for this conjecture is due to Gauthier, Le and Wollan, who recently proved that every graph G contains an immersion of a clique onvertices. Their result implies that every n-vertex graph with independence number α contains an immersion of a clique on n 3.54α − 1.13 vertices. We improve on this result for all α ≥ 3, by showing that every n-vertex graph with independence number α ≥ 3 contains an immersion of a clique on 4n 9(α−1) −⌊ α 2 ⌋ vertices.