A Local Ore-Type Condition for Graphs of Diameter Two to be Hamiltonian

Abstract

A graph is said to be in $L_1$ if $\mathrm{deg}(u)+\mathrm{deg}(v)\ge |N(u)\cup N(w)\cup N(v)|–1$ for each induced path $uwv$ of order three. We prove that a $2$-connected graph $G$ in $L_1$ of diameter two is hamiltonian, or $K_{d,d+1}\subset G\subset K_d+(d+1)K_1$ for some $d\ge 2$. This theorem generalizes a couple of known sufficient conditions for a graph to be hamiltonian. We also discuss the relation between this theorem and several other degree conditions for hamiltonicity.