Bipartite Kneser graphs are Hamiltonian

TL;DR

This paper proves that all bipartite Kneser graphs, except the Petersen graph, have a Hamilton cycle, confirming a long-standing conjecture and advancing understanding of their cycle structure.

Contribution

It establishes the Hamiltonicity of bipartite Kneser graphs $H(n,k)$ for all relevant parameters, confirming a major conjecture in graph theory.

Findings

Bipartite Kneser graphs $H(n,k)$ are Hamiltonian for all $n,k$ with $n eq 5,2$.

Almost all vertices in Kneser graphs $K(n,k)$ are visited by cycles when $n=2k+o(k)$.

Generalizes and improves previous results on cycle existence in Kneser 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 $[n]:=\{1,2,\ldots,n\}$ and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph $H(n,k)$ has as vertices all $k$-element and $(n-k)$-element subsets of $[n]$ and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all Kneser graphs and bipartite Kneser graphs except the Petersen graph $K(5,2)$ have a Hamilton cycle. The main contribution of this paper is proving this conjecture for bipartite Kneser graphs $H(n,k)$. We also establish the existence of cycles that visit almost all vertices in Kneser graphs $K(n,k)$ when $n=2k+o(k)$, generalizing and improving upon previous results on this problem.