The sum number of d-partite complete hypergraphs

• Autorzy: Hanns-Martin Teichert
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

sum number; sum hypergraphs; d-partite complete hypergraph

Kategorie tematyczne

• 05C78: Graph labelling (graceful graphs, bandwidth, etc.)
• 05C65: Hypergraphs

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 1999
• Tom: 19
• Numer: 1
• Strony: 79-91

Daty

wydano

1999

otrzymano

1998-08-24

poprawiono

1999-01-04

Twórcy

autor

Hanns-Martin Teichert

• Institute of Mathematics, Medical University of Lübeck, Wallstraße 40, 23560 Lübeck, Germany

Bibliografia

• [1] C. Berge, Hypergraphs, (North Holland, Amsterdam-New York-Oxford-Tokyo, 1989).
• [2] D. Bergstrand, F. Harary, K. Hodges. G. Jennings, L. Kuklinski and J. Wiener, The Sum Number of a Complete Graph, Bull. Malaysian Math. Soc. (Second Series) 12 (1989) 25-28.
• [3] Z. Chen, Harary's conjectures on integral sum graphs, Discrete Math. 160 (1996) 241-244, doi: 10.1016/0012-365X(95)00163-Q.
• [4] Z. Chen, Integral sum graphs from identification, Discrete Math. 181 (1998) 77-90, doi: 10.1016/S0012-365X(97)00046-0.
• [5] M.N. Ellingham, Sum graphs from trees, Ars Comb. 35 (1993) 335-349.
• [6] F. Harary, Sum Graphs and Difference Graphs, Congressus Numerantium 72 (1990) 101-108.
• [7] F. Harary, Sum Graphs over all the integers, Discrete Math. 124 (1994) 99-105, doi: 10.1016/0012-365X(92)00054-U.
• [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.
• [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.
• [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).

Identyfikatory

DOI

10.7151/dmgt.1087