<p>Fault tolerance is a crucial indicator of network stability. A reliable network maintains normal data transmission in the event of partial failures. The <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ n $</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-dimensional hypercube <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ Q_n $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is a well-known and efficient structure used in interconnection networks. In this study, we consider the Hamiltonian laceability of the hypercube <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Q_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> by extending a matching in the presence of disjoint faulty edges. Moreover, we prove that a matching <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ M $</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ Q_n $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ n \ge 6 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>, together with a matching <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ F $</EquationSource> <EquationSource Format="MATHML"><math> <mi>F</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ Q_n - M $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Q</mi> <mi>n</mi> </msub> <mo>-</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ |M\cup F| \le 3n-16 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>M</mi> <mo>∪</mo> <mi>F</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>3</mn> <mi>n</mi> <mo>-</mo> <mn>16</mn> </mrow> </math></EquationSource> </InlineEquation> can be extended to a Hamiltonian path between two vertices of opposite parity in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ Q_n - F $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Q</mi> <mi>n</mi> </msub> <mo>-</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation> that contains <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ M $</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation>.</p>

Hamiltonian laceability of hypercubes extending a set of matchings with disjoint faulty edges

期刊论文
发表日期：

2026年3月4日

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Abid Ali,
Gohar Ali,
Weihua Yang

摘要

Fault tolerance is a crucial indicator of network stability. A reliable network maintains normal data transmission in the event of partial failures. The $ n $ $n$ -dimensional hypercube $ Q_n $ $Q_{n}$ is a well-known and efficient structure used in interconnection networks. In this study, we consider the Hamiltonian laceability of the hypercube $Q_n$ $Q_{n}$ by extending a matching in the presence of disjoint faulty edges. Moreover, we prove that a matching $ M $ $M$ in $ Q_n $ $Q_{n}$ for $ n \ge 6 $ $n \geq 6$ , together with a matching $ F $ $F$ in $ Q_n - M $ $Q_{n} - M$ satisfying $ |M\cup F| \le 3n-16 $ $| M \cup F | \leq 3 n - 16$ can be extended to a Hamiltonian path between two vertices of opposite parity in $ Q_n - F $ $Q_{n} - F$ that contains $ M $ $M$ .