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The sum number of d-partite complete hypergraphs

Hanns-Martin Teichert — 1999

Discussiones Mathematicae Graph Theory

A d-uniform hypergraph is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that is isomorphic to the hypergraph d ( S ) = ( V , ) , where V = S and = v , . . . , v d : ( i j v i v j ) i = 1 d v i S . For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices w , . . . , w σ V such that w , . . . , w σ is a sum hypergraph. In this paper, we prove σ ( n , . . . , n d d ) = 1 + i = 1 d ( n i - 1 ) + m i n 0 , 1 / 2 ( i = 1 d - 1 ( n i - 1 ) - n d ) , where n , . . . , n d d denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.

Sum labellings of cycle hypergraphs

Hanns-Martin Teichert — 2000

Discussiones Mathematicae Graph Theory

A hypergraph is a sum hypergraph iff there are a finite S ⊆ IN⁺ and d̲, [d̅] ∈ IN⁺ with 1 < d̲ ≤ [d̅] such that is isomorphic to the hypergraph d ̲ , [ d ̅ ] ( S ) = ( V , ) where V = S and = e S : d ̲ | e | [ d ̅ ] v e v S . For an arbitrary hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices y , . . . , y σ V such that y , . . . , y σ is a sum hypergraph. Generalizing the graph Cₙ we obtain d-uniform hypergraphs where any d consecutive vertices of Cₙ form an edge. We determine sum numbers and investigate properties of sum labellings for this...

Classes of hypergraphs with sum number one

Hanns-Martin Teichert — 2000

Discussiones Mathematicae Graph Theory

A hypergraph ℋ is a sum hypergraph iff there are a finite S ⊆ ℕ⁺ and d̲,d̅ ∈ ℕ⁺ with 1 < d̲ < d̅ such that ℋ is isomorphic to the hypergraph d ̲ , d ̅ ( S ) = ( V , ) where V = S and = e S : d ̲ < | e | < d ̅ v e v S . For an arbitrary hypergraph ℋ the sum number(ℋ ) is defined to be the minimum number of isolatedvertices w , . . . , w σ V such that w , . . . , w σ is a sum hypergraph. For graphs it is known that cycles Cₙ and wheels Wₙ have sum numbersgreater than one. Generalizing these graphs we prove for the hypergraphs ₙ and ₙ that under a certain condition for the edgecardinalities...

Iterated neighborhood graphs

Martin SonntagHanns-Martin Teichert — 2012

Discussiones Mathematicae Graph Theory

The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph ( V , E N ) where E N = a,b | a ≠ b, x,a ∈ E and x,b ∈ E for some x ∈ V. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite. We present some results concerning the k-iterated neighborhood graph N k ( G ) : = N ( N ( . . . N ( G ) ) ) of G. In particular we investigate conditions for G and k such that N k ( G ) becomes a complete graph.

Competition hypergraphs of digraphs with certain properties II. Hamiltonicity

Martin SonntagHanns-Martin Teichert — 2008

Discussiones Mathematicae Graph Theory

If D = (V,A) is a digraph, its competition hypergraph (D) has vertex set V and e ⊆ V is an edge of (D) iff |e| ≥ 2 and there is a vertex v ∈ V, such that e = N D ( v ) = w V | ( w , v ) A . We give characterizations of (D) in case of hamiltonian digraphs D and, more general, of digraphs D having a τ-cycle factor. The results are closely related to the corresponding investigations for competition graphs in Fraughnaugh et al. [4] and Guichard [6].

Competition hypergraphs of digraphs with certain properties I. Strong connectedness

Martin SonntagHanns-Martin Teichert — 2008

Discussiones Mathematicae Graph Theory

If D = (V,A) is a digraph, its competition hypergraph 𝓒𝓗(D) has the vertex set V and e ⊆ V is an edge of 𝓒𝓗(D) iff |e| ≥ 2 and there is a vertex v ∈ V, such that e = {w ∈ V|(w,v) ∈ A}. We tackle the problem to minimize the number of strong components in D without changing the competition hypergraph 𝓒𝓗(D). The results are closely related to the corresponding investigations for competition graphs in Fraughnaugh et al. [3].

Products Of Digraphs And Their Competition Graphs

Martin SonntagHanns-Martin Teichert — 2016

Discussiones Mathematicae Graph Theory

If D = (V, A) is a digraph, its competition graph (with loops) CGl(D) has the vertex set V and {u, v} ⊆ V is an edge of CGl(D) if and only if there is a vertex w ∈ V such that (u, w), (v, w) ∈ A. In CGl(D), loops {v} are allowed only if v is the only predecessor of a certain vertex w ∈ V. For several products D1 ⚬ D2 of digraphs D1 and D2, we investigate the relations between the competition graphs of the factors D1, D2 and the competition graph of their product D1 ⚬ D2.

Niche Hypergraphs

Christian GarskeMartin SonntagHanns-Martin Teichert — 2016

Discussiones Mathematicae Graph Theory

If D = (V,A) is a digraph, its niche hypergraph NH(D) = (V, E) has the edge set ℇ = {e ⊆ V | |e| ≥ 2 ∧ ∃ v ∈ V : e = N−D(v) ∨ e = N+D(v)}. Niche hypergraphs generalize the well-known niche graphs (see [11]) and are closely related to competition hypergraphs (see [40]) as well as double competition hypergraphs (see [33]). We present several properties of niche hypergraphs of acyclic digraphs.

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