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

We say a graph G has a Hamiltonian path if it has a path containing all vertices of G. For a graph G , let σ 2 (G) denote the minimum degree sum of two nonadjacent vertices of G ; restrictions on σ 2 (G) are known as Ore-type conditions. Given an integer t ≥ 5 , we prove that if a connected graph G on n vertices satisfies σ 2 (G) > t − 3 t − 2 n , then G has either a Hamiltonian path or an induced subgraph isomorphic to K 1 , t. Moreover, we characterize all n -vertex graphs G where σ 2 (G) = t − 3 t − 2 n and G has neither a Hamiltonian path nor an induced subgraph isomorphic to K 1 , t. This is an analogue of a recent result by Momège (2018), who investigated the case when t = 4. [ABSTRACT FROM AUTHOR]