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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 ${⁺}_d\left(S\right)=(V,)$, where V = S and $=v₁,...,v_d:\left(i\ne j\Rightarrow v_i\ne v_j\right)\wedge {\sum }_{i=1}^d{v}_i\in S$. For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $w₁,...,w_{\sigma }\notin V$ such that $\cup w₁,...,w_{\sigma }$ is a sum hypergraph. In this paper, we prove $\sigma {(}_{n₁,...,n_d}^d)=1+{\sum }_{i=1}^d(n_i-1)+min0,\lceil 1/2({\sum }_{i=1}^{d-1}(n_i-1)-n_d)\rceil$, where ${}_{n₁,...,n_d}^d$ 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 ${}_{d̲,\left[d̅\right]}\left(S\right)=(V,)$ where V = S and $=e\subseteq S:d̲\le \left|e\right|\le \left[d̅\right]\wedge {\sum }_{v\in e}v\in S$. For an arbitrary hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $y₁,...,y_{\sigma }\notin V$ such that $\cup y₁,...,y_{\sigma }$ 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 ${\mathscr{H}}_{d̲,d̅}\left(S\right)=(V,)$ where V = S and $=e\subseteq S:d̲<\left|e\right|<d̅\wedge {\sum }_{v\in e}v\in S$. For an arbitrary hypergraph ℋ the sum number(ℋ ) is defined to be the minimum number of isolatedvertices $w₁,...,w_{\sigma }\notin V$ such that $\mathscr{H}\cup w₁,...,w_{\sigma }$ 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 $(V,E_N)$ 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 $N^k\left(G\right):=N\left(N(...N\left(G\right))\right)$ of G. In particular we investigate conditions for G and k such that $N^k\left(G\right)$ 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 $e=N_D⁻\left(v\right)=w\in V\left|\right(w,v)\in A$. 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].

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.