Extending two results on hamiltonian graphs involving the bipartite-hole-number
Kun Cheng, Yurui Tang
TL;DR
This paper advances Hamiltonicity criteria based on the bipartite-hole-number $\tilde\alpha(G)$ by proving an Ore-type condition: for a $2$-connected graph $G$ of order $n\ge3$, if $\sigma_2(G)\ge 2\tilde\alpha(G)-2$, then $G$ is Hamiltonian unless $n$ is odd and $G=G_{(n-1)/2}\vee \frac{n+1}{2}K_1$. It also establishes a stability version of the McDiarmid–Yolov result: if $\delta(G)\ge \tilde\alpha(G)-1$, then $G$ is Hamiltonian unless $G$ belongs to two explicitly described exceptional families. The authors’ approach relies on detailed structural analysis via Hamilton paths and the absence of $(s,t)$-bipartite-holes, yielding a sharp characterization of the non-Hamiltonian extremal graphs. Together, these results extend prior two-result frameworks (Li–Liu; Ellingham–Huang–Wei) and reinforce the role of bipartite-hole-number in governing Hamiltonicity, with immediate consequences for traceability and related graph-closure phenomena.
Abstract
The bipartite-hole-number of a graph $G$, denoted by $\widetildeα(G)$, 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, we first prove that any $2$-connected graph $G$ satisfying $d_G(x)+d_G(y)\ge 2\widetildeα(G)-2$ for every pair of non-adjacent vertices $x,y$ is hamiltonian except for a special family of graphs, thereby extending results of Li and Liu (2025), and Ellingham, Huang and Wei (2025). We then establish a stability version of a theorem by McDiarmid and Yolov (2017): every graph whose minimum degree is at least its bipartite-hole-number minus one is hamiltonian except for a special family of graphs.