Bipartite holes, degree sums and Hamilton cycles

The {\em bipartite-hole-number} of a graph $G$, denoted as $\widetilde\alpha(G)$, 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 $\delta(G) \ge \widetilde\alpha(G)$ is Hamiltonian. Later, Dragani\'c, Munh\'a Correia and Sudakov proved that $\delta\ge \widetilde\alpha(G)$ 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 $\sigma_2(G) \ge 2 \widetilde\alpha(G) - 1$, and that a connected graph $G$ contains a cycle through all vertices of degree at least $\widetilde\alpha(G)$. Both results extended McDiarmid and Yolov's result. As a step toward proving pancyclicity, we show that if an $n$-vertex graph $G$ satisfies $\sigma_2(G) \ge 2 \widetilde\alpha(G) - 1$, 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.