A sharp Ore-type condition for a connected graph with no induced star to have a Hamiltonian path

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We say a graph G has a Hamiltonian path if it has a path containing all vertices of G. For a graph G, let σ2(G) denote the minimum degree sum of two nonadjacent vertices of G; restrictions on σ2(G) are known as Ore-type conditions. Given an integer t≥5, we prove that if a connected graph G on n vertices satisfies σ2(G)>t−3t−2n, then G has either a Hamiltonian path or an induced subgraph isomorphic to K1,t. Moreover, we characterize all n-vertex graphs G where σ2(G)=t−3t−2n and G has neither a Hamiltonian path nor an induced subgraph isomorphic to K1,t. This is an analogue of a recent result by Momège (2018), who investigated the case when t=4.