Hadwiger's conjecture for the complements of Kneser graphs

TL;DR

This paper proves Hadwiger's conjecture for the complements of Kneser graphs, establishing that such graphs contain a complete minor of order equal to their chromatic number.

Contribution

The paper demonstrates that Hadwiger's conjecture holds for the complements of Kneser graphs, a significant class of graphs in combinatorics.

Findings

Hadwiger's conjecture verified for complements of Kneser graphs

Complements of Kneser graphs contain complete minors matching their chromatic number

Advances understanding of graph minors in specific graph classes

Abstract

Hadwiger's conjecture asserts that every graph with chromatic number $t$ contains a complete minor of order $t$. Given integers $n \ge 2k+1 \ge 5$, the Kneser graph $K(n, k)$ is the graph with vertices the $k$-subsets of an $n$-set such that two vertices are adjacent if and only if the corresponding $k$-subsets are disjoint. We prove that Hadwiger's conjecture is true for the complements of Kneser graphs.