### Some new sufficient conditions for 2p-Hamilton-biconnectedness of graphs

#### Abstract

A balanced bipartite graph $G$ is said to be \emph $2p$-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.

`` 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.

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