Kneser graphs are Hamiltonian

TL;DR

The paper resolves the long-standing conjecture by proving that every Kneser graph $K(n,k)$ with $n\geq 2k+3$ is Hamiltonian, with the Petersen graph as the sole exception, and extends the result to all connected generalized Johnson and generalized Kneser graphs. The authors introduce a constructive two-stage approach: first partition the vertex set into a cycle factor using a local parenthesis-matching map $f$, then join the resulting cycles via edge-disjoint 4-cycles guided by a glider-based analysis. The method hinges on a rich combinatorial encoding in terms of Motzkin paths, gliders with speeds summing to $k$, and a detailed dynamic accounting of overtakes, captured by a linear-motion framework to prove eventual movement of all gliders. Their technique not only settles the Hamiltonicity of these vertex-transitive families but also yields a constructive Gray-code style listing of $(n,k)$-combinations with disjointness adjacency, offering potential algorithmic applications. As a broader impact, the results substantially advance Lovász's conjecture for natural intersecting-set graph families and illuminate the structure of Hamilton cycles in highly symmetric combinatorial graphs.

Abstract

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph $K(5,2)$. This problem received considerable attention in the literature, including a recent solution for the sparsest case $n=2k+1$. The main contribution of this paper is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph $J(n,k,s)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two sets whose intersection has size exactly $s$. Clearly, we have $K(n,k)=J(n,k,0)$, i.e., generalized Johnson graph include Kneser graphs as a special case. Our results imply that all known natural families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász' conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway's Game of Life, and to analyze this system combinatorially and via linear algebra.

Figures (28)

• Figure 1: Relation between Hamiltonicity results established in this paper and previous papers. Arrows indicate implications.
• Figure 2: The Petersen graph $K(5,2)$. The vertices are all 2-elements subsets of $[5]=\{1,2,3,4,5\}$, and in the corresponding bitstrings, 1s are represented by black squares and 0s by white squares.
• Figure 3: Cycles of the factor ${\mathcal{C}}_{n,k}$ in several different Kneser graphs $K(n,k)$. The cycles in (a) and (b) are shown completely, whereas in (c) and (d) only the first 15 vertices are shown. 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 colored in the same color, with the opaque filling given to 1-bits, and the transparent filling given to 0-bits. (a) one glider of speed 1; (b) one glider of speed 2; (c) two gliders with speeds 1 and 2 that participate in an overtaking; (d) three gliders of speeds 1, 2 and 3 that participate in multiple overtakings. Animations of these examples are available at gliders.
• Figure 4: Gluing of two cycles from the factor via a 4-cycle in $K(13,5)$.
• Figure 5: Parenthesis matching for a bitstring $x\in X_{30,12}$. Matched pairs of bits are indicated by square brackets, and bits that are complemented to obtain $f(x)$ are highlighted.

Theorems & Definitions (74)

• Theorem 1
• Theorem 2
• Corollary 3
• Lemma 4
• proof
• Lemma 5: MR888679
• Lemma 6
• proof
• proof : Proof of Theorem \ref{['thm:Jnks']}
• Lemma 7