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The hypercube is one of the most popular interconnection networks since ithas simple structure and is easy to implement. The $n$-dimensional twistedcube, denoted by $TQ_n$, an important variation of the hypercube, possessessome properties superior to the hypercube. Recently, some interestingproperties of $TQ_n$ were investigated. In this paper, we construct twoedge-disjoint Hamiltonian cycles in $TQ_n$ for any odd integer $n\geqslant 5$.The presence of two edge-disjoint Hamiltonian cycles provides an advantage whenimplementing two algorithms that require a ring structure by allowing messagetraffic to be spread evenly across the twisted cube. Furthermore, we constructtwo equal node-disjoint cycles in $TQ_n$ for any odd integer $n\geqslant 3$, inwhich these two cycles contain the same number of nodes and every node appearsin one cycle exactly once. In other words, we decompose a twisted cube into twocomponents with the same size such that each component contains a Hamiltoniancycle.

Constructing Two Edge-Disjoint Hamiltonian Cycles and Two Equal Node-Disjoint Cycles in Twisted Cubes