Sparse Kneser Graphs Are Hamiltonian

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,, 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 + 2', k) with k >= 3 and a > 0 have a Hamilton cycle. We also prove that 6 K(2k + 1, k) has at least 2(2k-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.