Combination of Bases and an Evaluation of the Set of Extremal 3-Uniform Hypergraphs

Beretskii, I. S.; Egorova, E. K.; Mokryakov, A. V.; Tsurkov, V. I.

doi:10.1134/s1064230723050039

Cited by 2 publications

(6 citation statements)

References 15 publications

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“…The algorithm presented in [49] can be briefly described as an enumeration algorithm with the construction of lengthening chains of single-element bases that create a composite base. To form the base, a set of all combinations of index sets is pre-created: Ckn is the set of all combinations of k integers, from 1 to n, ordered in ascending order (Ckn is ordered in lexicographical order, for example, if n = 5 and k = 3; then, Ckn = {{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}}).…”

Section: An Approach To Calculating the Cardinality Of A Set Of Extre...mentioning

confidence: 99%

“…It is also important to note that the maximum number of elements in the base has so far only been estimated for the cases k = 2 and k = 3. In [49], the following estimates for the maximum base length are obtained MLB k n :…”

Section: An Approach To Calculating the Cardinality Of A Set Of Extre...mentioning

confidence: 99%

“…A number of fundamental works are devoted to the problems of estimating the power of graphs [42][43][44][45][46][47][48], but for hypergraphs such work has not been carried out in general; presumably, it is connected with the extremely indeterminate structure of hypergraphs, which led to a number of estimates in particular cases. In this way, the problems of enumerating hypergraphs and the problems of determining the power of a set only for certain classes of hypergraphs were solved [49][50][51][52][53].…”

Section: Introductionmentioning

confidence: 99%

“…In this work, the algorithm used in [49] will be analyzed, and its analogue, working on the basis of second-order signatures, will be presented.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Finding Set Extreme 3-Uniform Hypergraphs Cardinality through Second-Order Signatures

Egorova,

Leonov,

Mokryakov

et al. 2024

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This paper continues the study of second-order signature properties—the characterization of the extreme 3-uniform hypergraph. Previously, bases were used to count extreme 3-uniform hypergraphs. However, the algorithm using this mechanism is extremely labor-intensive. The structure of the signature allows us to use it as a more efficient basis for the same problem. Here, we establish the nature of the mutual correspondence between the kind of second-order signature and extreme hypergraphs, and we present a new algorithm to find the power of the set of extreme 3-uniform hypergraphs through the set of their characteristic-signatures. New results obtained with the proposed tool are also presented.

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Section: An Approach To Calculating the Cardinality Of A Set Of Extre...mentioning

confidence: 99%

Section: An Approach To Calculating the Cardinality Of A Set Of Extre...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In this work, the algorithm used in [49] will be analyzed, and its analogue, working on the basis of second-order signatures, will be presented.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finding Set Extreme 3-Uniform Hypergraphs Cardinality through Second-Order Signatures

Egorova,

Leonov,

Mokryakov

et al. 2024

Axioms

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show abstract

“…Their main advantage is that there is a one-to-one correspondence between the degree vectors of vertices and the extremal hypergraphs corresponding to them. Other compact characterizations, "base" [23] and "signature" [24], were obtained for the same class of hypergraphs. The latter is true only for the two-dimensional and three-dimensional cases.…”

Section: Introductionmentioning

confidence: 99%

The Algebra of Signatures for Extreme Two-Uniform Hypergraphs

Egorova,

Mokryakov,

Tsurkov

2023

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In the last decade, several characterizations have been constructed for constructions such as extreme hypergraphs. One of the most recently described features is the signature. A signature is a number that uniquely describes an extremal and allows one to efficiently store the extremal two-uniform hypergraph itself. However, for the signature, although various algorithms have been derived for transforming it into other object-characteristics such as the base, the adjacency matrix, and the vector of vertex degrees, no isolated signature union and intersection apparatus has been constructed. This allows us to build efficient algorithms based on signatures, the most compact representation of extremal two-uniform hypergraphs. The nature of the algebraic construction that can be built on a set of signatures using union and intersection operations has also been defined. It is proved that an algebra on a set of signatures with either the union or intersection operation forms a monoid; if the algebra is defined on a set of signatures with both union and intersection operations, it forms a distributive lattice.

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Combination of Bases and an Evaluation of the Set of Extremal 3-Uniform Hypergraphs

Cited by 2 publications

References 15 publications

Finding Set Extreme 3-Uniform Hypergraphs Cardinality through Second-Order Signatures

Finding Set Extreme 3-Uniform Hypergraphs Cardinality through Second-Order Signatures

The Algebra of Signatures for Extreme Two-Uniform Hypergraphs

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