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The Kneser graph K ( n , k ) has all k-subsets of an n-set as its vertices and two subsets are adjacent if they are disjoint. Lovasz conjectured that every connected vertex-transitive graph has a hamiltonian path. For n ⩾ 2 k + 1 , the Kneser graphs form a well-studied family of connected, regular, vertex-transitive graphs. A direct computation of hamiltonian cycles in K ( n , k ) is not feasible for large values of k, because K ( n , k ) has ( n k ) vertices. We give a sufficient condition for K ( 2 k + 2 , k ) to be hamiltonian for odd k: the existence of a particular hamiltonian path in a reduced graph over K ( 2 k + 2 , k ) . Also, we extend this result to the bipartite Kneser graphs B ( 2 k + 2 , k ) for odd k. |
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