Extending two results on hamiltonian graphs involving the bipartite-hole-number

TL;DR

This paper extends existing results on Hamiltonian graphs by relating bipartite-hole-number to degree conditions, providing new sufficient conditions for Hamiltonicity and a stability version of a related theorem.

Contribution

It generalizes previous Hamiltonian graph conditions involving bipartite-hole-number and establishes a stability result linking minimum degree and bipartite-hole-number.

Findings

Graphs satisfying certain degree sum conditions are Hamiltonian except for specific families.

A stability version shows minimum degree at least bipartite-hole-number minus one implies Hamiltonicity.

Extends results of Li, Liu, Ellingham, Huang, and Wei from 2025 and McDiarmid and Yolov from 2017.

Abstract

The bipartite-hole-number of a graph $G$, denoted by $\widetilde{\alpha}(G)$, 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 $|A|=s$ and $|B|=t$, there is an edge between $A$ and $B$. In this paper, we first prove that any $2$-connected graph $G$ satisfying $d_G(x)+d_G(y)\ge 2\widetilde{\alpha}(G)-2$ 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.