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1
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Sum labellings of cycle hypergraphs

100%
Teichert H.-M.
Discussiones Mathematicae Graph Theory
|
2000
|
tom 20
|
nr 2
255-265
EN
A hypergraph 𝓗 is a sum hypergraph iff there are a finite S ⊆ IN⁺ and d̲, [d̅] ∈ IN⁺ with 1 < d̲ ≤ [d̅] such that 𝓗 is isomorphic to the hypergraph $𝓗_{d̲,[d̅]} (S) = (V,𝓔)$ where V = S and $𝓔 = {e ⊆ S:d̲ ≤ |e| ≤ [d̅] ∧ ∑_{v∈ e} v ∈ S}$. For an arbitrary hypergraph 𝓗 the sum number σ = σ(𝓗) is defined to be the minimum number of isolated vertices $y₁,..., y_σ ∉ V$ such that $𝓗 ∪ {y₁,...,y_σ}$ 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 class of cycle hypergraphs.
2
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The sum number of d-partite complete hypergraphs

100%
Teichert H.-M.
Discussiones Mathematicae Graph Theory
|
1999
|
tom 19
|
nr 1
79-91
EN
A d-uniform hypergraph 𝓗 is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that 𝓗 is isomorphic to the hypergraph $𝓗 ⁺_d(S) = (V,𝓔)$, where V = S and $𝓔 = {{v₁,...,v_d}: (i ≠ j ⇒ v_i ≠ v_j)∧ ∑^d_{i=1} v_i ∈ S}$. For an arbitrary d-uniform hypergraph 𝓗 the sum number σ = σ(𝓗) is defined to be the minimum number of isolated vertices $w₁,...,w_σ ∉ V$ such that $𝓗 ∪{ w₁,..., w_σ}$ is a sum hypergraph. In this paper, we prove $σ(𝓚^{d}_{n₁,...,n_d}) = 1 + ∑^d_{i=1} (n_i -1 ) + min{0,⌈1/2(∑_{i=1}^{d-1} (n_i -1) - n_d)⌉}$, where $𝓚^{d}_{n₁,...,n_d}$ denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.
3
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Classes of hypergraphs with sum number one

100%
Teichert H.-M.
Discussiones Mathematicae Graph Theory
|
2000
|
tom 20
|
nr 1
93-103
EN
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 $ℋ_{d̲,d̅}(S) = (V,𝓔)$ where V = S and $𝓔 = {e ⊆ S: d̲ < |e| < d̅ ∧ ∑_{v∈ e} v∈ S}$. For an arbitrary hypergraph ℋ the sum number(ℋ ) is defined to be the minimum number of isolatedvertices $w₁,..., w_σ∉ V$ such that $ℋ ∪ {w₁,..., w_σ}$ 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 𝓒ₙ and 𝓦ₙ that under a certain condition for the edgecardinalities (𝓒ₙ)= (𝓦ₙ)=1
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Products Of Digraphs And Their Competition Graphs

64%
Sonntag M., Teichert H.-M.
Discussiones Mathematicae Graph Theory
|
2016
|
tom 36
|
nr 1
43-58
EN
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.
5
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Competition hypergraphs of digraphs with certain properties II. Hamiltonicity

64%
Sonntag M., Teichert H.-M.
Discussiones Mathematicae Graph Theory
|
2008
|
tom 28
|
nr 1
23-34
EN
If D = (V,A) is a digraph, its competition hypergraph 𝓒𝓗 (D) has vertex set V and e ⊆ V is an edge of 𝓒𝓗 (D) iff |e| ≥ 2 and there is a vertex v ∈ V, such that $e = N_D⁻(v) = {w ∈ V|(w,v) ∈ A}$. We give characterizations of 𝓒𝓗 (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].
6
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Competition hypergraphs of digraphs with certain properties I. Strong connectedness

64%
Sonntag M., Teichert H.-M.
Discussiones Mathematicae Graph Theory
|
2008
|
tom 28
|
nr 1
5-21
EN
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].
7
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Iterated neighborhood graphs

64%
Sonntag M., Teichert H.-M.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
|
nr 3
403-417
EN
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.
8
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Niche Hypergraphs

51%
Garske C., Sonntag M., Teichert H.-M.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
|
nr 4
819-832
EN
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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