Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube Networks

Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube Networks

In a network G, if each vertex of G is incident to at least <inline-formula> <tex-math notation="LaTeX">$g \, (\geq 1)$ </tex-math></inline-formula> fault-free vertices, then we say the network is g-conditionally faulty. An enhanced hypercube <inline-formula>...

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Bibliographic Details
Main Author: Min Liu
Format: Article
Language:English
Published: IEEE 2024-01-01
Series:IEEE Access
Subjects:
Cycles embedding
enhanced hypercube
conditionally faulty model
interconnection network
Online Access:https://ieeexplore.ieee.org/document/10550942/
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Summary:In a network G, if each vertex of G is incident to at least <inline-formula> <tex-math notation="LaTeX">$g \, (\geq 1)$ </tex-math></inline-formula> fault-free vertices, then we say the network is g-conditionally faulty. An enhanced hypercube <inline-formula> <tex-math notation="LaTeX">$Q_{n,k}$ </tex-math></inline-formula> is a network, which is an attractive variant of the hypercube <inline-formula> <tex-math notation="LaTeX">$Q_{n}$ </tex-math></inline-formula> by adding complementary edges between any vertices with the complementary addresses. Let <inline-formula> <tex-math notation="LaTeX">$F_{v}^{*}$ </tex-math></inline-formula> be the set of faulty vertices in <inline-formula> <tex-math notation="LaTeX">$Q_{n,k}$ </tex-math></inline-formula>. In this paper, in the 4-conditionally faulty <inline-formula> <tex-math notation="LaTeX">$Q_{n,k}$ </tex-math></inline-formula>, we show that <inline-formula> <tex-math notation="LaTeX">$Q_{n,k}-F_{v}^{*}$ </tex-math></inline-formula> contains a fault-free even cycle ranging in length from 4 to <inline-formula> <tex-math notation="LaTeX">$2^{n}-2|F_{v}^{*}|$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$n\geq 3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$|F_{v}^{*}|\leq 2n-4$ </tex-math></inline-formula>; and also contains a fault-free odd cycle ranging in length from <inline-formula> <tex-math notation="LaTeX">$n-k+2$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$2^{n}-2|F_{v}^{*}|-1$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$n \, (\geq 3)$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$2\nmid (n-k)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$|F_{v}^{*}|\leq 2n-5$ </tex-math></inline-formula>.
ISSN:2169-3536

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