Short proof that Kneser graphs are Hamiltonian for n ⩾ 4k

Abstract

Let n⩾k⩾1 be integers and define [n]={1,…,n}. For a set A let A(k) be the set of all k-element subsets of A. The Kneser graph K(n,k) is the graph with vertex set V=[n](k) and edge set E={{x,y}∈V(2):x∩y=∅}. Chen proved that for n⩾3k, Kneser graphs are Hamiltonian and later improved this to n⩾2.62k+1. Furthermore, Chen and Füredi gave a short proof that if k|n, Kneser graphs are Hamiltonian for n⩾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⩾4k.