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

Abstract

We say a graph $G$ has a Hamiltonian path if it has a path containing all vertices of $G$. For a graph $G$, let ${\sigma }_2(G)$ denote the minimum degree sum of two nonadjacent vertices of $G$; restrictions on ${\sigma }_2(G)$ are known as Ore-type conditions. Given an integer $t\ge 5$, we prove that if a connected graph $G$ on $n$ vertices satisfies ${\sigma }_2(G)>\frac{t-3}{t-2}n$, 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 ${\sigma }_2(G)=\frac{t-3}{t-2}n$ 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, who investigated the case when $t=4$.