The Kneser graph K(n, k) is the graph whose vertices are all the subsets with k elements of a set that has n elements, and two vertices are joined by an edge if the corresponding pair of k-sets is disjoint. The odd graph Ok is the special case of the Kneser graph when n=2k+1. A long-standing conjecture due to Lovász claims that Ok has a hamiltonian path for k ≥ 1. Previously, Lovász’s conjecture had been proved for all k ≤ 13. We have improved these avalues by showing that Ok has a hamiltonian path for 14 ≤ k ≤ 17. Additionally, we have established how close the odd graphs are to being hamiltonian: Ok has a closed spanning walk or trail in which every vertex appears at mosttwice.