DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 19(1) (1999) 79-91
DOI: 10.7151/dmgt.1087

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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[11] A. Sharary, Integral sum graphs from caterpillars, 1996 (to appear).
[12] M. Sonntag and H.-M. Teichert, The sum number of hypertrees, 1997 (to appear).
[13] M. Sonntag and H.-M. Teichert, On the sum number and integral sum number of hypertrees and complete hypergraphs, Proc. 3rd Kraków Conf. on Graph Theory, 1997 (to appear).

Received 24 August 1998
Revised 4 January 1999