Remarks on Partially Square Graphs, Hamiltonicity and Circumference

Given a graph G, its partially square graph G∗ is a graph obtained by adding an edge (u, v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition NG(x) ⊆ NG[u] ∪ NG[v], where NG[x] = NG(x) ∪ {x}. In the case where G is a claw-free graph, G∗ is equal to G. We define σ◦ t = min{ ∑ x∈S dG(x) : S is an independent set in G ∗ and |S| = t}. We give for hamiltonicity and circumference new sufficient conditions depending on σ◦ and we improve some known results.