Degree conditions restricted to induced paths for hamiltonicity of claw-heavy graphs

TL;DR

This paper explores specific degree conditions related to induced paths in 2-connected claw-heavy graphs, providing new criteria that ensure Hamiltonicity and improve upon previous results.

Contribution

It introduces Ore-type degree conditions restricted to induced P6 paths in 2-connected claw-heavy graphs, guaranteeing Hamiltonicity.

Findings

New degree conditions ensure Hamiltonicity in claw-heavy graphs.

Improves upon previous Hamiltonicity criteria for P6-free graphs.

Provides a broader class of Hamiltonian graphs under Ore-type conditions.

Abstract

Broersma and Veldman proved that every 2-connected claw-free and $P_6$-free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected claw-heavy and $P_6$-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and $P_6$-\emph{o}-heavy but not hamiltonian. In this paper we further give some Ore-type degree conditions restricting to induced $P_6$'s of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.