A Fan-type condition involving bipartite independence number for hamiltonicity in graphs

Abstract

The bipartite independence number of a graph $G$, denoted by $\tilde{\alpha }(G)$, is defined as the smallest integer $q$ for which there exist positive integers $s$ and $t$ with $s+t=q+1$, such that for any two disjoint subsets $A,B\subseteq V(G)$ with $|A|=s$ and $|B|=t$, there exists an edge between $A$ and $B$. In this paper, we prove that for a 2-connected graph $G$ of order at least three, if $\max \{d_G(x),d_G(y)\}\ge \tilde{\alpha }(G)$ for every pair of nonadjacent vertices $x,y$ at distance two, then $G$ is hamiltonian. Moreover, we prove that if $G$ is 3-connected and $\max \{d_G(x),d_G(y)\}\ge \tilde{\alpha }(G)+1$ for every pair of nonadjacent vertices $x,y$ at distance two, then $G$ is hamiltonian-connected. Our results generalize the recent work by Li and Liu.