# Clique immersions and independence number

## TL;DR

This paper improves bounds on clique immersions in graphs with given independence number, advancing the understanding of the analogue of Hadwiger's conjecture for the immersion order.

## Contribution

It provides a stronger bound for clique immersions in graphs with independence number at least 3, refining previous results.

## Key findings

- Improved lower bounds for clique immersions in graphs with independence number ≥ 3.
- Enhanced understanding of the relationship between independence number and clique immersion size.
- Generalization involving a nonnegative function f for the bounds.

## Abstract

The analogue of Hadwiger's conjecture for the immersion order states that every graph $G$ contains $K_{\chi (G)}$ as an immersion. If true, it would imply that every graph with $n$ vertices and independence number $\alpha$ contains $K_{\lceil \frac n\alpha\rceil}$ 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 $\bigl\lceil \frac{\chi (G)-4}{3.54}\bigr\rceil$ vertices. Their result implies that every $n$-vertex graph with independence number $\alpha$ contains an immersion of a clique on $\bigl\lceil \frac{n}{3.54\alpha}-1.13\bigr\rceil$ vertices. We improve on this result for all $\alpha\ge 3$, by showing that every $n$-vertex graph with independence number $\alpha\ge 3$ contains an immersion of a clique on $\bigl\lfloor \frac {n}{2.25 \alpha - f(\alpha)} \bigr\rfloor - 1$ vertices, where $f$ is a nonnegative function.