Sparse Kneser graphs are Hamiltonian

Abstract

For integers k > 1 and n > 2k+1, the Kneser graph K(n,k) is the graph whose vertices are the k-element subsets of {1, horizontal ellipsis ,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 > 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+2a,k) with k > 3 and a > 0 have a Hamilton cycle. We also prove that K(2k+1,k) has at least 22k-6 distinct Hamilton cycles for k > 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.