Hamiltonicity of Schrijver graphs and stable Kneser graphs

TL;DR

This paper proves that all Schrijver graphs and stable Kneser graphs have Hamilton cycles, which can be efficiently computed, advancing understanding of their combinatorial and algorithmic properties.

Contribution

It establishes the existence of Hamilton cycles in all stable Kneser graphs, including Schrijver graphs, with an efficient algorithm for their construction.

Findings

All $S(n,k,s)$ graphs admit Hamilton cycles.

Hamilton cycles can be computed in linear time per vertex.

Results apply to Schrijver graphs as a special case.

Abstract

For integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $S(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More generally, for integers $k\geq 1$, $s\geq 2$, and $n \geq sk+1$, 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 $\mathcal{O}(n)$ per generated vertex.