Degree condition on distance three for Hamiltonian-biconnected in balanced bipartite graphs

Let G be a connected balanced bipartite graph of order 2n with $n\ge 2$. For $x,y\in V\left(G\right)$, we denote by dist $(x,y)$ the distance between x and y, that is, the length of a shortest path connecting x and y. Denote ${\mu }_3\left(G\right)=\min \left\{d\right(x)+d(y):\text{dist}(x,y)=3,x,y\in V(G\left)\right\}$, if G is not a complete bipartite graph; otherwise ${\mu }_3\left(G\right)=+\infty$. In 1993, Amar proved that if ${\mu }_3\left(G\right)\ge n+1$, then G is hamiltonian. In this paper, we first prove that ${\mu }_3\left(G\right)\ge n$ implies that G contains a hamiltonian path. Using this result, we also characterize all non-hamiltonian bipartite graphs with ${\mu }_3\left(G\right)=n$. Moreover, we prove that if ${\mu }_3\left(G\right)\ge n+2$ or G is 3-connected and ${\mu }_3\left(G\right)\ge n+1$, then G is hamiltonian-biconnected. The lower bounds provided in these results are shown to be sharp.