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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

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.

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1 Discussiones Mathematicae Graph Theory

1 Teichert H.-M.

1 1999

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The sum number of d-partite complete hypergraphs