# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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# Discussiones Mathematicae Graph Theory

## THE SUM NUMBER OF d-PARTITE COMPLETE HYPERGRAPHS

Hanns-Martin Teichert

Institute of Mathematics, Medical University of Lübeck
Wallstraß e 40, 23560 Lübeck, Germany

## Abstract

A d-uniform hypergraph H is a sum hypergraph iff there is a finite S ⊆ [.2em][l]IN+ such that H is isomorphic to the hypergraph Hd+(S) = (V,E), where V = S and E = {{ v1,…,vd}:(i ≠ j⇒ vi ≠ vj)∧∑di = 1 vi ∈ 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,…,wσ∉ V such that H∪{ w1,…, wσ} is a sum hypergraph.

In this paper, we prove

 σ(Kdn1,…,nd) = 1 + d ∑ i = 1 (ni-1) + min ⎧ ⎨ ⎝ 0, ⎡ ⎢ ⎢ 1 -- 2 ⎛ ⎝ d-1 ∑ i = 1 (ni-1)-nd ⎞ ⎠ ⎤ ⎥ ⎥ ⎫> ⎬ ⎭ ,

where Kdn1,…,nd denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.

Keywords: sum number, sum hypergraphs, d-partite complete hypergraph.

1991 Mathematics Subject Classification: 05C65, 05C78.

## References

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