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A d-uniform hypergraph is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that is isomorphic to the hypergraph , where V = S and . For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices such that is a sum hypergraph.
In this paper, we prove
,
where denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.
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 where V = S and . For an arbitrary hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices such that 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...
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 where V = S and . For an arbitrary hypergraph ℋ the sum number(ℋ ) is defined to be the minimum number of isolatedvertices such that 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...
The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph where = 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 of G. In particular we investigate conditions for G and k such that becomes a complete graph.
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 . 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].
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].
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.
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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