Variations on a sufficient condition for Hamiltonian graphs

Given a 2-connected graph G on n vertices, let G∗ be its partially square graph, obtained by adding edges uv whenever the vertices u, v have a common neighbor x satisfying the condition NG(x) ⊆ NG[u] ∪ NG[v], where NG[x] = NG(x) ∪ {x}. In particular, this condition is satisfied if x does not center a claw (an induced K1,3). Clearly G ⊆ G∗ ⊆ G, where G is the square of G. For any independent triple X = {x, y, z} we define σ3(X) = d(x) + d(y) + d(z)− |N(x) ∩N(y) ∩N(z)| . Flandrin et al. proved that a 2-connected graph G is hamiltonian if σ3(X) ≥ n holds for any independent triple X in G. Replacing X in G by X in the larger graph G∗, Wu et al. improved recently this result. In this paper we characterize the nonhamiltonian 2-connected graphs G satisfying the condition σ3(X) ≥ n− 1 where X is independent in G∗. Using the concept of dual closure we (i) give a short proof of the above results and (ii) we show that each graph G satisfying this condition is hamiltonian if and only if its dual closure does not belong to two well defined exceptional classes of graphs. This implies that it takes a polynomial time to check the nonhamiltonicity or the hamiltonicity of such G.