Short proof that Kneser graphs are Hamiltonian for $n\geq 4k$

TL;DR

This paper provides a concise proof demonstrating that Kneser graphs are Hamiltonian for all parameters where n is at least four times k, extending previous results that required larger n relative to k.

Contribution

The paper introduces a short proof establishing Hamiltonicity of Kneser graphs for n ≥ 4k without divisibility constraints, improving upon earlier bounds.

Findings

Kneser graphs are Hamiltonian for n ≥ 4k.

Previous bounds required larger n, such as n ≥ 3k or n ≥ 2.62k+1.

The new proof simplifies the understanding of Hamiltonian cycles in Kneser graphs.

Abstract

For integers $n\geq k\geq 1$, the Kneser graph $K(n,k)$ is the graph with vertex set $V=[n]^{(k)}$ and edge set $E=\{\{x,y\} \in V^{(2)}: x\cap y=\emptyset\}$. Chen proved that for $n\geq 3k$, Kneser graphs are Hamiltonian and later improved this to $n\geq 2.62k+1$. Furthermore, Chen and F\"uredi gave a short proof that if $k | n$, Kneser graphs are Hamiltonian for $n\geq 3k$. In this note, we present a short proof that does not need the divisibility condition, i.e., we give a short proof that $K(n,k)$ is Hamiltonian for $n\geq 4k$.