Hamiltonicity of Schrijver graphs and stable Kneser graphs

Abstract

For integers k≥1 and n≥2k+1, the Schrijver graph S(n,k) has as vertices all k-element subsets of [n]≔{1,2,…,n} that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More generally, for integers k≥1, s≥2, and n≥sk+1, the s-stable Kneser graph S(n,k,s) has as vertices all k-element subsets of [n] in which any two elements are in cyclical distance at least s. We prove that all the graphs S(n,k,s), in particular Schrijver graphs S(n,k)=S(n,k,2), admit a Hamilton cycle that can be computed in time O(n) per generated vertex.