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5 pages, 212 KiB

Open AccessArticle

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 1187

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

\bar{χ}

(H) = n - s

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

Open AccessArticle

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 3 | Viewed by 1459

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 \leq 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

Open AccessArticle

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 1686

Abstract

Given a homogeneous ideal

I \subseteq k [x_{0}, \dots, x_{n}]

, the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs

m, r \in N

[...] Read more.

Given a homogeneous ideal

I \subseteq k [x_{0}, \dots, x_{n}]

, the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs

m, r \in N

I^{(m)} \subseteq I^{r}

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

P_{k}^{n}

. 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

Open AccessArticle

Uniformly Resolvable Decompositions of K_v-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 2 | Viewed by 1751

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 \in Γ

. In this article we completely solve the existence problem for decompositions of

K_{v}

-I into

C_{n}

-factors and

K_{1, 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

Open AccessArticle

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 3 | Viewed by 1560

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

χ_{s} (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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