Partially Square Graphs, Hamiltonicity and Circumference II

Given a graph G, its partially square graph G∗ is a graph obtained by adding an edge uv 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 case G is a claw-free graph, G∗ is equal to G2, We define σ∗t = min{∑x∈ d∗G(x): S is an independent set in G∗ and ∣S∣ = t}, where d∗G(x) = ∣{y ∈ V∣ xy ∈ E(G∗)}∣. e for hamiltonicity and circumference new sufficient conditions depending on and we improve some known results.