Opuscula Math. 25, no. 2 (2005), 319-323

 
Opuscula Mathematica

A note on self-complementary 4-uniform hypergraphs

Artur Szymański

Abstract. We prove that a permutation \(\theta\) is complementing permutation for a \(4\)-uniform hypergraph if and only if one of the following cases is satisfied: (i) the length of every cycle of \(\theta\) is a multiple of \(8\), (ii) \(\theta\) has \(1\), \(2\) or \(3\) fixed points, and all other cycles have length a multiple of \(8\), (iii) \(\theta\) has \(1\) cycle of length \(2\), and all other cycles have length a multiple of \(8\), (iv) \(\theta\) has \(1\) fixed point, \(1\) cycle of length \(2\), and all other cycles have length a multiple of \(8\), (v) \(\theta\) has \(1\) cycle of length \(3\), and all other cycles have length a multiple of \(8\). Moreover, we present algorithms for generating every possible \(3\) and \(4\)-uniform self-complementary hypergraphs.

Keywords: complementing permutation, self-complementary hypergraph, \(k\)-uniform hypergraph.

Mathematics Subject Classification: 34A30, 34B05, 35B12, 35A15, 35J50, 35J65, 46T15, 34B15.

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  • Artur Szymański
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
  • Received: 2004-12-23.
Opuscula Mathematica - cover

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Artur Szymański, A note on self-complementary 4-uniform hypergraphs, Opuscula Math. 25, no. 2 (2005), 319-323

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