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

TL;DR

This paper establishes a sharp Ore-type degree condition for connected graphs to contain a Hamiltonian path or an induced star, extending previous results and characterizing extremal cases.

Contribution

It introduces a new Ore-type condition involving $\sigma_2(G)$ that guarantees Hamiltonian paths or induced stars, generalizing earlier work for specific cases.

Findings

Proves that $\sigma_2(G) > rac{t-3}{t-2}n$ ensures a Hamiltonian path or an induced $K_{1,t}$.

Characterizes extremal graphs where $\sigma_2(G) = rac{t-3}{t-2}n$ without Hamiltonian paths or induced stars.

Extends and generalizes recent results by Momège for the case $t=4$.

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 $t\geq 5$, we prove that if a connected graph $G$ on $n$ vertices satisfies $\sigma_2(G)>{t-3\over t-2}n$, 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 $\sigma_2(G)={t-3\over t-2}n$ 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 Mom\`ege, who investigated the case when $t=4$.