Bipartite holes, degree sums and Hamilton cycles
Mark Ellingham, Yixuan Huang, Bing Wei
TL;DR
This work extends Dirac-type Hamiltonicity results via the bipartite-hole-number $\widetilde{\alpha}(G)$ by establishing a degree-sum framework using $\sigma_2(G)$ and relating structural extremals to $K_k\oplus_1 K_{n+1-k}$. It proves that $\delta(G) \ge \widetilde{\alpha}(G)$ guarantees Hamiltonicity (and, under $2$-connectivity, that $\sigma_2(G) \ge 2\widetilde{\alpha}(G)-1$ suffices for Hamiltonicity), while also showing the existence of a cycle through all vertices of degree at least $\widetilde{\alpha}(G)$ (X-cyclability). A key consequence is that, under the same bound, either $G$ contains a triangle or $G$ is a balanced complete bipartite graph $K_{n/2,n/2}$, linking to pancyclicity directions. The paper also derives intrinsic relations between connectivity and $\widetilde{\alpha}(G)$, notably $\kappa(G)+\widetilde{\alpha}(G) \le n$, and discusses how these results interact with Bondy–Chvátal closure, including infinite families where closure-based arguments do not apply. Overall, the work broadens the Dirac/ Ore paradigm with a bipartite-hole parameter, clarifying the landscape of Hamiltonicity, cyclability, and connectivity.
Abstract
The {\em bipartite-hole-number} of a graph $G$, denoted as $\widetildeα(G)$, is the minimum number $k$ such that there exist integers $a$ and $b$ with $a + b = k+1$ such that for any two disjoint sets $A, B \subseteq V(G)$, there is an edge between $A$ and $B$. McDiarmid and Yolov initiated research on bipartite holes by extending Dirac's classical theorem on minimum degree and Hamiltonian cycles. They showed that a graph on at least three vertices with $δ(G) \ge \widetildeα(G)$ is Hamiltonian. Later, Draganić, Munhá Correia and Sudakov proved that $δ\ge \widetildeα(G)$ implies that $G$ is pancyclic, unless $G = K_{\frac n2, \frac n2}$. This extended the result of McDiarmid and Yolov and generalized a theorem of Bondy on pancyclicity. In this paper, we show that a $2$-connected graph $G$ is Hamiltonian if $σ_2(G) \ge 2 \widetildeα(G) - 1$, and that a connected graph $G$ contains a cycle through all vertices of degree at least $\widetildeα(G)$. Both results extended McDiarmid and Yolov's result. As a step toward proving pancyclicity, we show that if an $n$-vertex graph $G$ satisfies $σ_2(G) \ge 2 \widetildeα(G) - 1$, then it either contains a triangle or it is $K_{\frac n2, \frac n2}$. Finally, we discuss the relationship between connectivity and the bipartite hole number.