An Ore-type condition for hamiltonicity in graphs

The bipartite-hole-number of a graph $G$, denoted as $\tilde{\alpha }(G)$, is the minimum number $k$ such that there exist positive integers $s$ and $t$ with $s+t=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$ with $\mathrm{\mid }A\mathrm{\mid }=s$ and $\mathrm{\mid }B\mathrm{\mid }=t$, there is an edge between $A$ and $B$. In this paper, based on Ore-type conditions, we show that if a graph $G$ is 2-connected and the degree sum of any two nonadjacent vertices in $G$ is at least $2\tilde{\alpha }(G)$, then $G$ is hamiltonian. Furthermore, we prove that if $G$ is 3-connected and the degree sum of any two nonadjacent vertices in $G$ is at least $2\tilde{\alpha }(G)+1$, then $G$ is hamiltonian-connected.