Sparse Kneser graphs are Hamiltonian

For integers k ≥1 and n ≥2 k +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 (2 k +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 (2 k +1, k ) has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form K (2 k +2 a , k ) with k ≥3 and a ≥0 have a Hamilton cycle. We also prove that K (2 k +1, k ) has at least 2 2 k −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.

Authors

• Torsten Mütze
• Jerri Nummenpalo
• Bartosz Walczak

Keywords

• Hamilton cycle
• Kneser graph
• odd graph
• vertex-transitive graph