2017/11/30 by Torsten Mütze, Jerri Nummenpalo, Bartosz Walczak · 1 citation
Computer Science · Engineering · Mathematics · #1-planar graph #Chordal graph #Combinatorics #Complete graph #Discrete mathematics #Disjoint sets #Graph #Hamiltonian path #Hypergraph #Limits and Structures in Graph Theory #Mathematics #Odd graph #Topological and Geometric Data Analysis #cs.DM #graph theory and CDMA systems #math.CO #msc:05C45 #msc:94B25
paper · pdf · doi:10.1112/jlms.12406
published as J. London Math. Soc. 103 (2021) 1253-1275
arxiv created 2020/09/30 · openalex publication_date 2020/12/10 · arxiv updated 2021/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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+2a,k) with k≥ 3 and a≥ 0 have a Hamilton cycle. We also prove that 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.