Opuscula Math. 25, no. 2 (2005), 319-323
Opuscula Mathematica
A note on self-complementary 4-uniform hypergraphs
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.
- Artur Szymański
- AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
- Received: 2004-12-23.
