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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2024-01-01
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| description | 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>. |
| format | Article |
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| spelling | doaj-art-f11dbc7f9b874bb5a05e93825d0d03c22025-08-20T02:40:13ZengIEEEIEEE Access2169-35362024-01-0112800628007010.1109/ACCESS.2024.341017510550942Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube NetworksMin Liu0https://orcid.org/0000-0003-1317-8935School of Statistics and Data Science, Ningbo University of Technology, Ningbo, Zhejiang, ChinaIn 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>.https://ieeexplore.ieee.org/document/10550942/Cycles embeddingenhanced hypercubeconditionally faulty modelinterconnection network |
| spellingShingle | Min Liu Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube Networks IEEE Access Cycles embedding enhanced hypercube conditionally faulty model interconnection network |
| title | Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube Networks |
| title_full | Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube Networks |
| title_fullStr | Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube Networks |
| title_full_unstemmed | Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube Networks |
| title_short | Vertex-Fault-Tolerant Cycles Embedding in Four-Conditionally Faulty Enhanced Hypercube Networks |
| title_sort | vertex fault tolerant cycles embedding in four conditionally faulty enhanced hypercube networks |
| topic | Cycles embedding enhanced hypercube conditionally faulty model interconnection network |
| url | https://ieeexplore.ieee.org/document/10550942/ |
| work_keys_str_mv | AT minliu vertexfaulttolerantcyclesembeddinginfourconditionallyfaultyenhancedhypercubenetworks |