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Advances in Design Theory and Applications in Combinatorial Algebraic Geometry
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Advances in Design Theory and Applications in Combinatorial Algebraic Geometry

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "B: Geometry and Topology".

Deadline for manuscript submissions: closed (30 September 2021) | Viewed by 11948

Special Issue Editors

Prof. Dr. Elena Guardo

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, University of Catania, Viale A. Doria, 6, 95100 Catania, Italy
Interests: commutative algebra; algebraic geometry, multilinar algebra; computer algebra; combinatorics; graph theory
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Mario Gionfriddo

E-Mail Website
Guest Editor
Dipartimento di Matematica e Informatica, Università di Catania, 95125 Catania, Italy
Interests: combinatorics; graph theory; hypergraphs; block design theory; Steiner systems; number theory; hypergroups
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Salvatore Milici

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, University of Catania, Catania, Italy
Interests: resolvable decompositions; combinatorics; graph theory; hypergraphs; block design theory; Steiner systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Combinatorial algebraic geometry is a branch of mathematics studying objects that can be interpreted from a combinatorial point of view (such as matroids, polytopes, codes or finite geometries) and also algebraically (using tools from group theory, lattice theory or commutative algebra), and which has applications in designs, coding theory, cryptography, and number theory.

This Special Issue on “Advances in Design Theory and applications in Combinatorial Algebraic Geometry” invites front-line researchers and authors to submit original research and review articles on exploring new trends in design theory. It is intended as a bridge between computational issues in the treatment of curves and surfaces (from the symbolic and also numeric points of view) and a combinatorial point of view.

Potential topics include but are not limited to:

  • Algebraic graph theory;
  • Finite geometry and designs;
  • Combinatorial algebraic geometry and its applications;
  • Combinatorial algebra and its applications;
  • Coding theory;
  • Statistical design and experiments;
  • Cryptography;
  • Number theory.

Keywords

  • Designs
  • Resolvable decompositions
  • Steiner Systems
  • Hilbert Functions
  • Fat points
  • Symbolic and regular powers of ideals: Waldschmidt constant and resurgence

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • Reprint: MDPI Books provides the opportunity to republish successful Special Issues in book format, both online and in print.

Further information on MDPI's Special Issue policies can be found here.

Published Papers (5 papers)

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5 pages, 212 KiB  
Article
Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs
by Nicholas Newman and Vitaly Voloshin
Mathematics 2021, 9(8), 828; https://doi.org/10.3390/math9080828 - 10 Apr 2021
Cited by 1 | Viewed by 1669
Abstract
In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph H that can be placed on a circle and colored according to some rules that are related to colorings of circular mixed hypergraphs. A [...] Read more.
In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph H that can be placed on a circle and colored according to some rules that are related to colorings of circular mixed hypergraphs. A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph of this cycle. We propose an algorithm to color the (r,r)-uniform, complete, circular, mixed hypergraphs for all feasible values with no gaps. In doing so, we show χ(H)=2 and χ¯(H)=n−s or n−s−1 where s is the sieve number. Full article
(This article belongs to the Special Issue Advances in Design Theory and Applications in Combinatorial Algebraic Geometry)
59 pages, 1973 KiB  
Article
Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs
by Anita Keszler and Zsolt Tuza
Mathematics 2021, 9(5), 484; https://doi.org/10.3390/math9050484 - 26 Feb 2021
Cited by 5 | Viewed by 2083
Abstract
In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C(r,k,v) of order v is a collection of r-uniform k-cycles on a v-element [...] Read more.
In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C(r,k,v) of order v is a collection of r-uniform k-cycles on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C(3,5,v) of orders v=25,26,31,35,37,41,46,47,55,56, a highly symmetric construction for v=40, and cyclic 2-split constructions of orders 32,40,50,52. As a consequence, all orders v≤60 permitted by the divisibility conditions admit a C(3,5,v) system. New recursive constructions are also introduced. Full article
(This article belongs to the Special Issue Advances in Design Theory and Applications in Combinatorial Algebraic Geometry)
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15 pages, 446 KiB  
Article
Steiner Configurations Ideals: Containment and Colouring
by Edoardo Ballico, Giuseppe Favacchio, Elena Guardo, Lorenzo Milazzo and Abu Chackalamannil Thomas
Mathematics 2021, 9(3), 210; https://doi.org/10.3390/math9030210 - 21 Jan 2021
Cited by 3 | Viewed by 2340
Abstract
Given a homogeneous ideal I⊆k[x0,…,xn], the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs m,r∈N [...] Read more.
Given a homogeneous ideal I⊆k[x0,…,xn], the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs m,r∈N, I(m)⊆Ir holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne–Huneke Conjecture, and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in Pkn. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly’s Conjectures. Moreover, given a hypergraph H, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to H. We apply these results in the case that H is a Steiner System. Full article
(This article belongs to the Special Issue Advances in Design Theory and Applications in Combinatorial Algebraic Geometry)
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9 pages, 769 KiB  
Article
Uniformly Resolvable Decompositions of Kv-I into n-Cycles and n-Stars, for Even n
by Giovanni Lo Faro, Salvatore Milici and Antoinette Tripodi
Mathematics 2020, 8(10), 1755; https://doi.org/10.3390/math8101755 - 13 Oct 2020
Cited by 3 | Viewed by 2326
Abstract
If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of [...] Read more.
If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X∈Γ. In this article we completely solve the existence problem for decompositions of Kv-I into Cn-factors and K1,n-factors in the case when n is even. Full article
(This article belongs to the Special Issue Advances in Design Theory and Applications in Combinatorial Algebraic Geometry)
10 pages, 320 KiB  
Article
Edge Balanced 3-Uniform Hypergraph Designs
by Paola Bonacini, Mario Gionfriddo and Lucia Marino
Mathematics 2020, 8(8), 1353; https://doi.org/10.3390/math8081353 - 12 Aug 2020
Cited by 6 | Viewed by 2111
Abstract
In this paper, we completely determine the spectrum of edge balanced H-designs, where H is a 3-uniform hypergraph with 2 or 3 edges, such that H has strong chromatic number χs(H)=3. Full article
(This article belongs to the Special Issue Advances in Design Theory and Applications in Combinatorial Algebraic Geometry)
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