Sparse Kneser graphs are Hamiltonian

TL;DR

This paper proves that all odd Kneser graphs with parameters $k extgreater{}=3$ contain Hamilton cycles, resolving a decades-old problem, and extends results to a broader class of Kneser graphs.

Contribution

It establishes the Hamiltonicity of all odd Kneser graphs for $k extgreater{}=3$, solving an old open problem and providing new lower bounds on the number of Hamilton cycles.

Findings

All odd Kneser graphs $K(2k+1,k)$ with $k extgreater{}=3$ are Hamiltonian.

At least $2^{2^{k-6}}$ Hamilton cycles exist in these graphs for $k extgreater{}=6$.

The proof reduces Hamiltonicity to finding a spanning tree in a hypergraph on Dyck words.

Abstract

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k\geq 3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form $K(2k+2^a,k)$ with $k\geq 3$ and $a\geq 0$ have a Hamilton cycle. We also prove that $K(2k+1,k)$ has at least $2^{2^{k-6}}$ distinct Hamilton cycles for $k\geq 6$. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words.