Hamiltonian paths in odd graphs

Abstract

3Lovász conjectured that every connected vertex-transitive graph has a Hamiltonian path. The odd graphs O k form a well-studied family of connected, k-regular, vertex-transitive graphs. It was previously known that O k has Hamiltonian paths for k ≤ 14. A direct computation of Hamiltonian paths in O k is not feasible for large values of k, because O k has 2k − 1 k − 1 vertices and k 2 2k − 1 k − 1 edges. We show that O k has Hamiltonian paths for 15 ≤ k ≤ 18. Instead of directly running any heuristics, we use existing results on the middle levels problem, therefore further relating these two fundamental problems, namely finding a Hamiltonian path in the odd graph and finding a Hamiltonian cycle in the corresponding middle levels graph. We show that further improved results for the middle levels problem can be used to find Hamiltonian paths in O k for larger values of k.

On hamiltonian cycles in the prism over the odd graphs

“…The vertex set of B(n, k) can be seen as two (symmetric) layers of the n-dimensional cube. Indeed, if the n-dimensional cube is drawn so that, for each k, all sets in [n] addition, in [2] it has been shown that O k has a hamiltonian path for k ≤ 17. Kierstead and Trotter [12] and Duffus et al [5] found two different 1-factorizations of B k hoping that the union of two suitable 1-factors would provide a hamiltonian cycle of B k .…”