  The hypercube is one of the most popular interconnection networks since it
has simple structure and is easy to implement. The $n$-dimensional twisted
cube, denoted by $TQ_n$, an important variation of the hypercube, possesses
some properties superior to the hypercube. Recently, some interesting
properties of $TQ_n$ were investigated. In this paper, we construct two
edge-disjoint Hamiltonian cycles in $TQ_n$ for any odd integer $n\geqslant 5$.
The presence of two edge-disjoint Hamiltonian cycles provides an advantage when
implementing two algorithms that require a ring structure by allowing message
traffic to be spread evenly across the twisted cube. Furthermore, we construct
two equal node-disjoint cycles in $TQ_n$ for any odd integer $n\geqslant 3$, in
which these two cycles contain the same number of nodes and every node appears
in one cycle exactly once. In other words, we decompose a twisted cube into two
components with the same size such that each component contains a Hamiltonian
cycle.
