A category-theoretical approach to hypergraphs

Dörfler, Willibald; Waller, Derek A.

doi:10.1007/bf01224952

Cited by 38 publications

(35 citation statements)

References 8 publications

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“…For i = 1, 2, the projection π i is defined by π i : (v 1 , v 2 ) → v i . The categorical product of two hypergraphs H 1 and H 2 , defined by Dörfler and Waller in 1980 [10], is the hypergraph H 1 × H 2 with vertex set V 1 × V 2 and edge set…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Coloring properties of categorical product of general Kneser hypergraphs

2018

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More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture was generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier, in 2016, introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful type results for the coloring of categorical product of general Kneser hypergraphs, which generalize the Hajiabolhassan-Meunier result. Also, we present a new lower bound for the chromatic number of categorical product of general Kneser hypergraphs which can be much better than the Hajiabolhassan-Meunier lower bound. Using this lower bound, we enrich the family of hypergraphs satisfying Zhu's conjecture.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Coloring properties of categorical product of general Kneser hypergraphs

2018

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“…The associated graph G(HI) of 7"N is the graph with vertex set V = IV with two vertices in G(7H) adjacent if 7 has an edge containing both vertices. (It is interesting to note that G is a covariant functor from the category of hypergraphs to the category of graphs [3] and (S, n Si+,)g(S, n S,+,) and (Sj n Sj+,)g(Si n Si+,)(i j ). Finally, 7H is 3-acyclic if it has no Graham cycle [4].…”

Section: Definitionsmentioning

confidence: 99%

Interval Graphs and Hypergraph Acyclicity Degree

Parks¹

1991

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“…rsG X := rG (X,rs(X)) , rhG X := rG (X,rh(X)) ). 4 The category of reflexive oriented X-graphs (resp. reflexive symmetric X-graphs, reflexive hereditary X-graphs) is its category of presheaves roG X (resp.…”

Section: Introductionmentioning

confidence: 99%

Exponential Objects in Categories of Generalized Uniform Hypergraphs

Schmidt

2018

JAMCS

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We construct exponential objects in categories of generalized uniform hypergraphs and use embeddings induced by nerve-realization adjunctions to show why conventional categories of graphs and hypergraphs do not have exponential objects.As a consequence of our constructions we address the problem that the category of k-uniform hypergraphs (as defined in [4]) lacks connected colimits, exponentials and does not continuously embed into the category of hypergraphs. We prove there is a continuous embedding of the category of k-uniform hypergraphs in a category of (X, M)-graphs (Proposition 5.5) which preserves any relevant categorical structures (e.g., colimits, exponentials, injectives, projectives). Therefore, working in a category of (reflexive) (X, M)-graphs provides a better categorical environment for constructions on uniform hypergraphs. (X, M )-GraphsWe begin with a definition.Definition 2.1.1. Let M be a monoid and X a right M-set. The theory for (X, M)-graphs, G (X,M ) , is the category with two objects V and A and homsets given byComposition is defined as m • x = x.m (the right-action via M), m • m ′ = m ′ m (monoid operation of M). 2. Let M be a monoid such that the set Fix(M) := { m ′ ∈ M | ∀m ∈ M, m ′ m = m ′ } is non-empty. Let X := { x m ′ | m ′ ∈ Fix(M) } be the right M-set with rightaction x m .m ′ := x mm ′ for each m ∈ M and x m ′ ∈ X. The theory for reflexive (X, M)-graphs, rG (X,M ) is the same as for G (X,M ) but with rG (X,M ) (A, V ) := {ℓ}, and composition ℓ • m = ℓ, ℓ • x m ′ = id V , and x • ℓ = x for each m ∈ M and m ′ ∈ Fix(M).

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A category-theoretical approach to hypergraphs

Cited by 38 publications

References 8 publications

Coloring properties of categorical product of general Kneser hypergraphs

Coloring properties of categorical product of general Kneser hypergraphs

Interval Graphs and Hypergraph Acyclicity Degree

Exponential Objects in Categories of Generalized Uniform Hypergraphs

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