Abstract
The odd graph O k is the graph whose vertices are all subsets with k elements of a set {1,…,2k + 1}, and two vertices are joined by an edge if the corresponding pair of k-subsets is disjoint. A conjecture due to Biggs claims that O k is hamiltonian for k ≥ 3 and a conjecture due to Lovász implies that O k has a hamiltonian path for k ≥ 1. In this paper, we show that the prism over O k is hamiltonian and that O k has a cycle with .625|V(O k )| vertices at least.
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