  A balanced bipartite graph $G$ is said to be $2p$-Hamilton-biconnected if for
any balanced subset $W$ of size $2p$ of $V(G)$, the subgraph induced by
$V(G)\backslash W$ is Hamilton-biconnected. In this paper, we prove that "Let
$p\geq0$ and $G$ be a balanced bipartite graph of order $2n$ with minimum
degree $\delta(G)\geq k$, where $n\geq 2k-p+2$ and $k\geq p$. If the number of
edges $ e(G)>n(n-k+p-1)+(k+2)(k-p+1), $ then $G$ is $2p$-Hamilton-biconnected
except some exceptions." Furthermore, this result is used to present two new
spectral conditions for a graph to $2p$-Hamilton-biconnected. Moreover, the
similar results are also presented for nearly balanced bipartite graphs.
