Fault-Free Hamiltonian Cycles Passing through Prescribed Edges in <formula formulatype="inline"><tex Notation="TeX">$k$</tex> <mathgraphic graphicformat="GIF" fileref="zhang-ieq1-2311794.gif"/></formula>-Ary <formula formulatype="inline"><tex Notation="TeX"> $n$</tex><mathgraphic graphicformat="GIF" fileref="zhang-ieq2-2311794.gif"/></formula>-Cubes with Faulty Edges

Abstract

The $k$ -ary $n$ -cube $Q^k_n$ is one of the most attractive interconnection networks for parallel and distributed systems. In this paper, we consider the problem of a fault-free hamiltonian cycle passing through prescribed edges in a $k$ -ary $n$ -cube $Q^k_n$ with some faulty edges. The following result is obtained: For any $n\ge 2$ and $k\ge 3$ , let $F\subset E(Q^k_n)$ , ${\cal P}\subset E(Q^k_n)\setminus F$ with $\vert {\cal P}\vert \le 2n-2$ , $\vert F\vert \le 2n-(\vert {\cal P}\vert +2)$ . Then there exists a hamiltonian cycle passing through all edges of ${\cal P}$ in $Q^k_n-F$ if and only if the subgraph induced by ${\cal P}$ consists of pairwise vertex-disjoint paths. It improves the result given by Yang and Wang .