Introduction}
The hypercube network is one of the most popular interconnection networks. The balanced hypercube, proposed by Wu and Huang \cite{Huang2}, is one of the variations of the hypercube. It is a bipartite graph, node-transitive \cite{Huang2} and edge-transitive \cite{J.-X.Zhou}, but has smaller diameter than hypercubes and supports an efficient reconfiguration without changing the adjacent relationship among tasks \cite{Huang2}. In recent years, the balanced hypercube has attracted much attention in the literature \cite{Dongqin}, \cite{R.-X.Hao}, \cite{M.Xu}, \cite{M.-C.Yang}, \cite{J.-X.Zhou}.

One of the most central issues in the study of interconnection networks is to find parallel paths, which is naturally related to routing among nodes and fault tolerance of the network \cite{L.-H.Hsu},\cite{J.A.M.}. Parallel paths correspond to disjoint paths of the graph. The problem of node-disjoint paths has received much attention because of its numerous applications in high performance communication networks, fault-tolerant routings, and so on.

A graph is denoted by $G=(V,E)$, where $V$ is the node set and $E$ is the edge set of $G$. Two nodes $u$ and $v$ are adjacent if $(u,v)\in E$. A path $P[v_0,v_n]$ denoted by $\langle v_0, v_1, \ldots, v_n \rangle$, is a sequence of distinct nodes, where $v_i$ and $v_{i+1}$ are adjacent $(0\leq i\leq n)$. A path $P[v_0, v_n]$ also can be denoted by $\langle v_0, v_1, \ldots, v_i, P[v_i,v_j], v_j, v_{j+1},\ldots v_n \rangle$, where $P[v_i,v_j]=\langle v_i, v_{i+1}, \ldots, v_j \rangle$. A cycle $C$ is denoted by $\langle v_0, v_1, P[v_1, v_{n-1}],v_0 \rangle$, where $n\geq 3$. A {\it Hamiltonian cycle} (respectively, {\it Hamiltonian path}) of $G$ is a cycle (respectively, path) that traverses every node of $G$ exactly once. A path (resp. cycle) contains $k$ nodes is a {\it $k$-path} (resp. $k$-cycle). A graph is {\it bipartite} if its node sets can be partitioned into two disjoint subsets and each edge with two end nodes from different subsets. A graph G is said to be {\it Hamiltonian connected} if there exists a Hamiltonian path between any two vertices of $G$. It is easy to see that any bipartite graph with at least three vertices is not Hamiltonian connected. For this reason, Simmons \cite{Simmons} introduced the concept of Hamiltonian laceable for Hamiltonian bipartite graphs. A Hamiltonian bipartite graph is {\it Hamiltonian laceable} if there is a Hamiltonian path between any two vertices in different bipartite sets. Obviously, the Hamilton cycle can be embedded in the Hamiltonian connected graphs.
We say that $k$ paths are {\it disjoint} in a network if any two of them have no common node. For an embedding of linear arrays in a network, a cover implies that every node can participate in pipeline computation \cite{Park1}. A $k$-disjoint path cover ($k$-{\it DPC} for short) of a graph is a set of $k$ (internally) disjoint paths that altogether cover every node of the graph.
The problem of disjoint path covers arose from applications of interconnection networks in which the full utilization of nodes is important.  The disjoint path cover problem finds applications in many areas such as software testing, database design, and code optimization \cite{Asdre}, \cite{Ntafos}. We follow \cite{Bondy} for terminologies and notations not specified in the paper.

Let $S$ be a set of $k$ sources $S=\{s_{1},s_{2},\ldots,s_{k}\}$ and a set of $k$ sinks $T=\{t_{1},t_{2},\ldots,t_{k}\}$ in a graph $G$ such that $S\cap T=\emptyset$. A {\it (many-to-many)} $k$-{\it disjoint path cover} of $G$ relate to $S\cup T$ is a set of $k$ disjoint paths joining sources and sinks. It is called {\it paired} if each source $s_{i}$ is joined to a specified sink $t_{i}$. Otherwise, it is called {\it unpaired}.  A {\it paired} (resp. {\it unpaired}) $k$-{\it disjoint path cover} of a bipartite graph $G$ with two distinct partite set $B$ and $W$ is a paired (resp. unpaired) $k$-disjoint path cover of $G$ relate to $S\cup T$ for any two sets $S=\{s_{1},s_{2},\ldots,s_{k}\} \subseteq B$ and $T=\{t_{1},t_{2},\ldots,t_{k}\}\subseteq W$.
 Indeed, if $k=1$, then a paired $1$-disjoint path cover of a network is just a Hamiltonian path connecting the given two nodes. There are some results about $k$-disjoint path cover for $k=1$. However, for $k\geq 2$, the problem of $k$-disjoint path covers of networks is much more difficult. Research attention is focused on some special classes of well-known networks, especially bipartite graphs.
 As examples, for unpaired many-to-many disjoint path cover, Chen, Liu et al. and Park studied a class of bipartite graphs, hypercubes and hypercube-like interconnection networks in \cite{Xie-BinChen4}, \cite{Liu} and \cite{Park3} respectively. Chen, Jo et al. and Lai et al. discussed paired 2-disjoint path covers of hypercubes, bipartite hypercube-like graphs and matching composition network in
 \cite{Xie-BinChen1}, \cite{S.Jo} and \cite{Lai} respectively.

Since the $n$-dimensional balanced hypercube $BH_n$ is a bipartite graph, in the following, its two distinct partite sets are denoted by $B$ and $W$. (In fact, we regard that all vertices in $B$ are given a black color, all vertices in $W$ are given a white color). The paired 1-disjoint path cover of $BH_{n}$, which is the Hamiltonian laceability, was obtained by Xu et al. in \cite{M.Xu}; the paired 2-disjoint path cover of $BH_{n}$ was obtained by Cheng et al. in \cite{Dongqin}. In this paper, we get the paired 3-disjoint path cover of $BH_n$ with $n\geq 3$ which improves the above known results about the paired $k$-disjoint path covers of $BH_{n}$ for $k=1,2$.

\begin{theorem}\label{3-DPC}
Let $BH_{n}$ be the $n$-dimensional balanced hypercube for $n\geq 3$, let $S=\{s_{1}, s_{2}, s_{3}\}\subseteq B$ and $T=\{t_{1}, t_{2}, t_{3}\}\subseteq W$, then $BH_{n}$ has a paired 3-disjoint path cover relative to $S\cup T$, where $B$ and $W$ are two distinct partite sets of $BH_{n}$.
\end{theorem}

The rest of this paper is organized as follows. Section 2 introduces definitions and preliminaries of the balanced hypercube. The proof of our main result is given in Section 3. Section 4 concludes this paper.


\section{Definitions and preliminaries}
In this section, we give definitions and basic properties of the balanced hypercube.

An $n$-dimensional balanced hypercube \cite{Huang2}, denoted by $BH_{n}$, is defined as follows. In the following, $"+"$ and $"-"$ always mean an operation with mod 4. For convenience, ``mod 4" is omitted.

\begin{defi}\label{defi-1}
For $n\geq1$, $BH_n $ consists of $2^{2n}$ nodes $(a_0, a_1, \ldots,a_{i-1}, a_i, a_{i+1}, \ldots, a_{n-1})$, where $a_i\in\{0,1,2,3\}$ $(0\leq i\leq n-1)$. Every node $(a_0,a_1,\ldots,a_{i-1}, a_i, a_{i+1},\ldots, a_{n-1})$ connects the following $2n$ nodes:
\begin{enumerate}
  \item [{\rm (1)}] $(a_0\pm1, a_1, \ldots, a_{i-