# The sum number of d-partite complete hypergraphs

Discussiones Mathematicae Graph Theory (1999)

- Volume: 19, Issue: 1, page 79-91
- ISSN: 2083-5892

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topHanns-Martin Teichert. "The sum number of d-partite complete hypergraphs." Discussiones Mathematicae Graph Theory 19.1 (1999): 79-91. <http://eudml.org/doc/270567>.

@article{Hanns1999,

abstract = {A d-uniform hypergraph is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that is isomorphic to the hypergraph $ ⁺_d(S) = (V,)$, where V = S and $ = \{\{v₁,...,v_d\}: (i ≠ j ⇒ v_i ≠ v_j)∧ ∑^d_\{i=1\} v_i ∈ S\}$. For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $w₁,...,w_σ ∉ V$ such that $ ∪\{ w₁,..., w_σ\}$ is a sum hypergraph.
In this paper, we prove
$σ(^\{d\}_\{n₁,...,n_d\}) = 1 + ∑^d_\{i=1\} (n_i -1 ) + min\{0,⌈1/2(∑_\{i=1\}^\{d-1\} (n_i -1) - n_d)⌉\}$,
where $^\{d\}_\{n₁,...,n_d\}$ denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.},

author = {Hanns-Martin Teichert},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {sum number; sum hypergraphs; d-partite complete hypergraph; sum graph; labelling; hypergraph},

language = {eng},

number = {1},

pages = {79-91},

title = {The sum number of d-partite complete hypergraphs},

url = {http://eudml.org/doc/270567},

volume = {19},

year = {1999},

}

TY - JOUR

AU - Hanns-Martin Teichert

TI - The sum number of d-partite complete hypergraphs

JO - Discussiones Mathematicae Graph Theory

PY - 1999

VL - 19

IS - 1

SP - 79

EP - 91

AB - A d-uniform hypergraph is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that is isomorphic to the hypergraph $ ⁺_d(S) = (V,)$, where V = S and $ = {{v₁,...,v_d}: (i ≠ j ⇒ v_i ≠ v_j)∧ ∑^d_{i=1} v_i ∈ S}$. For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $w₁,...,w_σ ∉ V$ such that $ ∪{ w₁,..., w_σ}$ is a sum hypergraph.
In this paper, we prove
$σ(^{d}_{n₁,...,n_d}) = 1 + ∑^d_{i=1} (n_i -1 ) + min{0,⌈1/2(∑_{i=1}^{d-1} (n_i -1) - n_d)⌉}$,
where $^{d}_{n₁,...,n_d}$ denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.

LA - eng

KW - sum number; sum hypergraphs; d-partite complete hypergraph; sum graph; labelling; hypergraph

UR - http://eudml.org/doc/270567

ER -

## References

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- [8] N. Hartsfield and W.F. Smyth, The Sum Number of Complete Bipartite Graphs, in: R. Rees, ed., Graphs and Matrices (Marcel Dekker, New York, 1992) 205-211. Zbl0791.05090
- [9] M. Miller, J. Ryan, Slamin, Integral sum numbers of ${H}_{2,n}$ and ${K}_{m,m}$, 1997 (to appear).
- [10] A. Sharary, Integral sum graphs from complete graphs, cycles and wheels, Arab. Gulf J. Sci. Res. 14 (1) (1996) 1-14. Zbl0856.05088
- [11] A. Sharary, Integral sum graphs from caterpillars, 1996 (to appear). Zbl0856.05088
- [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).

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