<p>We initiate the study of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-fold near-factorizations of groups with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. While <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-fold near-factorizations of groups with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> have been studied in numerous papers, this is the first detailed treatment for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\lambda > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We establish fundamental properties of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-fold near-factorizations and introduce the notion of equivalence. We prove various necessary conditions of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-fold near-factorizations, including upper bounds on <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. We present three constructions of infinite families of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-fold near-factorizations, highlighting the characterization of two subfamilies of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-fold near-factorizations. We discuss a computational approach to <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-fold near-factorizations and tabulate computational results for abelian groups of small order.</p>

$\lambda $-fold near-factorizations of groups

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2026年5月12日

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Donald L. Kreher,
Shuxing Li,
Douglas R. Stinson

摘要

We initiate the study of $\lambda $ $λ$ -fold near-factorizations of groups with $\lambda > 1$ $λ > 1$ . While $\lambda $ $λ$ -fold near-factorizations of groups with $\lambda = 1$ $λ = 1$ have been studied in numerous papers, this is the first detailed treatment for $\lambda > 1$ $λ > 1$ . We establish fundamental properties of $\lambda $ $λ$ -fold near-factorizations and introduce the notion of equivalence. We prove various necessary conditions of $\lambda $ $λ$ -fold near-factorizations, including upper bounds on $\lambda $ $λ$ . We present three constructions of infinite families of $\lambda $ $λ$ -fold near-factorizations, highlighting the characterization of two subfamilies of $\lambda $ $λ$ -fold near-factorizations. We discuss a computational approach to $\lambda $ $λ$ -fold near-factorizations and tabulate computational results for abelian groups of small order.

\(\lambda \)-fold near-factorizations of groups

摘要