Kneser graphs are Hamiltonian

TL;DR

This paper proves that all Kneser graphs, except the Petersen graph, contain a Hamilton cycle, confirming a long-standing conjecture and extending the result to generalized Johnson graphs, using innovative combinatorial and algebraic methods.

Contribution

It establishes the Hamiltonicity of all Kneser graphs and connected generalized Johnson graphs, resolving a conjecture from the 1970s with novel kinetic system techniques.

Findings

All Kneser graphs (except Petersen) are Hamiltonian.

Extended Hamiltonicity to all connected generalized Johnson graphs.

Introduced a kinetic glider system approach for cycle analysis.

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 Lov\'asz' 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.