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 26(2) (2006) 217-224
DOI: 10.7151/dmgt.1314

SELF-COMPLEMENTARY HYPERGRAPHS

A. Paweł Wojda

AGH University of Science and Technology
Faculty of Applied Mathematics
Department of Discrete Mathematics
Al. Mickiewicza 30, 30-059 Kraków, Poland
e-mail: wojda@uci.agh.edu.pl

Abstract

A k-uniform hypergraph H = (V;E) is called self-complementary if there is a permutation σ:V→ V, called self-complementing, such that for every k-subset e of V, e ∈ E if and only if σ(e) ∉ E. In other words, H is isomorphic with
H′ = (V; ( V )−E).
k

In the present paper, for every k, (1 ≤ k ≤ n), we give a characterization of self-complementig permutations of k-uniform self-complementary hypergraphs of the order n. This characterization implies the well known results for self-complementing permutations of graphs, given independently in the years 1962-1963 by Sachs and Ringel, and those obtained for 3-uniform hypergraphs by Kocay, for 4-uniform hypergraphs by Szymański, and for general (not uniform) hypergraphs by Zwonek.

Keywords: k-uniform hypergraph, self-complementary hypergraph.

2000 Mathematics Subject Classification: 05C65.

References

[1] A. Benhocine and A.P. Wojda, On self-complementation, J. Graph Theory 8 (1985) 335-341, doi: 10.1002/jgt.3190090305.
[2] W. Kocay, Reconstructing graphs as subsumed graphs of hypergraphs, and some self-complementary triple systems, Graphs and Combinatorics 8 (1992) 259-276, doi: 10.1007/BF02349963.
[3] G. Ringel, Selbstkomplementäre Graphen, Arch. Math. 14 (1963) 354-358, doi: 10.1007/BF01234967.
[4] H. Sachs, Über selbstkomplementäre Graphen, Publ. Math. Debrecen 9 (1962) 270-288.
[5] A. Szymański, A note on self-complementary 4-uniform hypergraphs, Opuscula Mathematica 25/2 (2005) 319-323.
[6] M. Zwonek, A note on self-complementary hypergraphs, Opuscula Mathematica 25/2 (2005) 351-354.

Received 18 July 2005
Revised 4 February 2006