Better Approximations of Non-Hamiltonian Graphs

Although there have been a lot of efforts to seek nice characterization of non-Hamiltonian graphs, little progress has been made so far. An important progress was achieved by Chvital [5, 61 who introduced the class of non-l-tough graphs (Nl T) and the class of non-sub-2-factor graphs (NS2F). Both contain only non-Hamiltonian graphs and the conditions for membership can be checked in non-deterministic polynomial time. Also it is known that N 1 TG NS2F. Chvital posed an open question about the complexity of those classes, i.e., whether or not they are NP-complete [6]. Recently, Bauer et al. [2] proved that NlT is NP-complete. In this paper we prove: (i) NS2F is also NP-complete. (ii) NS2F NlT is DP-complete, namely, the former characterization for non-Hamiltonian graphs is essentially better than the latter. (iii) Those results are still true for the bounded-degree graphs.