Generalized group divisible t-designs, which form a common generalization of group divisible t-designs and generalized t-designs, were defined by Liu et al. (J Combin Des 31:575–603, 2023). In this paper, we define a related class of combinatorial designs which simultaneously provide a generalization of both group divisible covering designs and generalized covering designs. Referring to the research on generalized covering designs given by Bailey et al., we investigate the basic properties and derive several bounds on the minimum size of generalized group divisible covering designs. Moreover, we basically determine the existence of optimal generalized group divisible covering designs with index <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lambda =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, strength <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and block size <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|{\textbf {k}}|=3, 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="bold">k</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, and strength <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and block size <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|{\textbf {k}}|=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="bold">k</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, except possibly for the case of block type <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\textbf {k}}= (2,1,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">k</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Generalized group divisible covering designs

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发表日期：

2026年6月3日

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Jialu Wang,
Lijun Ma,
Lidong Wang,
Zihong Tian

摘要

Generalized group divisible t-designs, which form a common generalization of group divisible t-designs and generalized t-designs, were defined by Liu et al. (J Combin Des 31:575–603, 2023). In this paper, we define a related class of combinatorial designs which simultaneously provide a generalization of both group divisible covering designs and generalized covering designs. Referring to the research on generalized covering designs given by Bailey et al., we investigate the basic properties and derive several bounds on the minimum size of generalized group divisible covering designs. Moreover, we basically determine the existence of optimal generalized group divisible covering designs with index $\lambda =1$ $λ = 1$ , strength $t=2$ $t = 2$ and block size $|{\textbf {k}}|=3, 4$ $| k | = 3, 4$ , and strength $t=3$ $t = 3$ and block size $|{\textbf {k}}|=4$ $| k | = 4$ , except possibly for the case of block type ${\textbf {k}}= (2,1,1)$ $k = (2, 1, 1)$ .