Self-complementing permutations of k-uniform hypergraphs

Abstract : A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962).
For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd.
In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 <= s <= 2(l) and the following two conditions hold:
(i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and
(ii) Sigma(i:lambda(vertical bar Oi vertical bar)<= l) vertical bar O-i vertical bar <= r.
For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.
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Discrete Mathematics and Theoretical Computer Science, DMTCS, 2009, 11 (1), pp.117--123
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Artur Szymański, Adam Pawel Wojda. Self-complementing permutations of k-uniform hypergraphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2009, 11 (1), pp.117--123. 〈hal-00988180〉

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