Complete minors and stability numbers

TL;DR

This paper proves a new upper bound on the order of graphs based on their stability and Hadwiger numbers, advancing understanding related to Hadwiger's conjecture.

Contribution

It establishes a novel upper bound on graph order involving stability and Hadwiger numbers for graphs with stability number at least 3 and Hadwiger number at least 5.

Findings

Proves that n ≤ (α-1)(2h-5)+5 for specified graphs

Improves bounds related to Hadwiger's conjecture

Combines ideas from Kawarabayashi et al. and Wood

Abstract

Hadwiger's conjecture implies that $n\le\alpha h$ for all graphs of order $n$, stability number $\alpha $, and Hadwiger number $h$. Combining ideas of Kawarabayashi et al. and Wood, we prove that $n\le (\alpha-1)(2h-5)+5$ for such graphs if $\alpha\ge 3$ and $h\ge 5$.