Hamiltonicity of Schrijver graphs and stable Kneser graphs

TL;DR

The paper proves that Schrijver graphs $S(n,k)$ and, more generally, $s$-stable Kneser graphs $S(n,k,s)$ with $n\ge sk+1$ admit Hamilton cycles. It develops a two-step framework: (i) a cycle-factor construction by encoding vertices as bitstrings and forming cycles $C(x)$, and (ii) gluing these cycles together via carefully chosen connectors to obtain a single Hamilton cycle, with a parallel auxiliary graph ensuring global connectivity. An efficient algorithm is provided that, given a current vertex, computes the next vertex on the Hamilton cycle in $\mathcal{O}(n)$ time, and the approach extends to all $s\ge 2$ with only minor modifications. The work also connects to independent results by Ledezma and Pastine and offers practical implementation details, including an available C++ implementation. The findings advance understanding of Hamiltonicity in these graph families and yield concrete, scalable algorithms for generating Hamilton cycles.

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.

Figures (5)

• Figure 1: A cycle in the Schrijver graph $S(15,6)$ constructed from the mapping $\sigma$ (top), and a cycle in the Kneser graph $K(15,6)$ constructed from the more general mapping $f$. The cycle in $S(15,6)$ is shown completely, whereas only the first 15 vertices of the cycle in $K(15,6)$ are displayed. On the left, each vertex is represented by a bitstring, with 1-bits colored black and 0-bits colored white. The vertices are printed from top to bottom in the order of the cycle. The right-hand side shows the interpretation of certain groups of bits as gliders, and their movement over time. Matched bits belonging to the same glider are given the same color, with the opaque filling given to 1-bits, and the transparent filling given to 0-bits.
• Figure 2: Illustration of the proof of Lemma \ref{['lem:HU']}.
• Figure 3: Illustration of the tree ${\mathcal{T}}_p$ for $n=17$ and $k=7$. Each node of ${\mathcal{H}}_{n,k}[{\mathcal{T}}_p]$, i.e., each cycle $C(x)\in {\mathcal{C}}_{n,k}$ is represented by the string $L(\langle{x}\rangle)$, corresponding to the lexicographically least connectable vertex on that cycle. The edges are oriented to point towards the lexicographically smaller string. Horizontal and vertical edges correspond to the first and second case in \ref{['eq:Ptau']}, respectively.
• Figure 4: Illustration of the instructions (i)--(iv).
• Figure 5: Hamilton cycles in different $s$-stable Kneser graphs computed by the algorithm from Section \ref{['sec:proof-algo']}.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm:ham']} for Schrijver graphs $S(n,k)=S(n,k,2)$
• Lemma 5
• proof