Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube Networks
Min Liu · IEEE Access · 2024
In a network G, if each vertex of G is incident to at least$g \, (\geq 1)$fault-free vertices, then we say the network is g-conditionally faulty. An enhanced hypercube$Q_{n,k}$is a network, which is an attractive variant of the hypercube$Q_{n}$by adding complementary edges between any vertices with the complementary addresses. Let$F_{v}^{*}$be the set of faulty vertices in$Q_{n,k}$. In this paper, in the 4-conditionally faulty$Q_{n,k}$, we show that$Q_{n,k}-F_{v}^{*}$contains a fault-free even cycle ranging in length from 4 to$2^{n}-2|F_{v}^{*}|$, where$n\geq 3$and$|F_{v}^{*}|\leq 2n-4$; and also contains a fault-free odd cycle ranging in length from$n-k+2$to$2^{n}-2|F_{v}^{*}|-1$, where$n \, (\geq 3)$,$2 mid (n-k)$and$|F_{v}^{*}|\leq 2n-5$.