Clique Immersions and Independence Number

The analogue of Hadwiger’s conjecture for the immersion order states that every graph G contains K χ ( G ) as an immersion. If true, this 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 on ⌈ χ ( G ) − 4 3 . 54 ⌉ vertices. Their result implies that every n-vertex graph with independence number α contains an immersion of a clique on ⌈ n 3 . 54 α − 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 ⌊ n 2 . 25 α − f ( α ) ⌋ − 1 vertices, where f is a nonnegative function.