An Ore-type condition for hamiltonicity in graphs
Chengli Li, Feng Liu
TL;DR
The paper introduces the bipartite-hole-number $\widetilde{\alpha}(G)$ and ties it to Ore-type degree-sum conditions via $\sigma_2(G)$. It proves that a 2-connected graph with $\sigma_2(G) \ge 2\widetilde{\alpha}(G)$ is Hamiltonian, a corollary yields traceability when the bound is weakened to $\sigma_2(G) \ge 2\widetilde{\alpha}(G)-2$, and a 3-connected graph with $\sigma_2(G) \ge 2\widetilde{\alpha}(G)+1$ is Hamiltonian-connected. The paper also discusses the necessity of the connectivity hypotheses through explicit counterexamples and provides the standard join-based reduction in the corollary. Together, these results extend classical Ore-type theorems by incorporating bipartite-hole structure and connectivity.
Abstract
The bipartite-hole-number of a graph $G$, denoted as $\widetildeα(G)$, 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 $|A|=s$ and $|B|=t$, 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\widetildeα(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\widetildeα(G)+1$, then $G$ is hamiltonian-connected.