Generating self-complementary uniform hypergraphs

Abstract

In 2007, Szyma«ski and Wojda proved that for positive integers n, k with k ≤ n, a self-complementary k-uniform hypergraph of order n exists if and only if n k is even. In this paper, we characterize the cycle type of a k-complementing permutation in Sym(n) which has order equal to a power of 2. This yields a test to determine if a nite permutation is a k-complementing permutation, and an algorithm for generating all selfcomplementary k-hypergraphs of order n, up to isomorphism, for feasible n. We also obtain an alternative description of the necessary and sucient conditions on the order of a self-complementary k-uniform hypergraph, in terms of the binary representation of k. This extends previous results for the cases k = 2, 3, 4 due to Ringel, Sachs, Suprunenko, Kocay and Szyma«ski.

Edge‐transitive homogeneous factorizations of complete uniform hypergraphs

“…Let be as in (I) of Lemma 5. Then = 3, and we may choose = 8 and = (1,8). Recall that every is the union of some -orbits on {3} .…”

“…Then ≤ . It is well-known that = GL( , 2) acts primitively on the set of 2-dimensional subspaces of 2 8) is permutation isomorphic to a 3-homogeneous subgroup of AGL(3, 2) with (1,8) corresponding to (3,2). It follows that { | 1 ≤ ≤ 7} is isomorphic to  (8;3,7) .…”

“…Self-complementary uniform hypergraphs have been extensively studied, see [14,16,[18][19][20]25] and the references therein for self-complementary graphs, and see [7][8][9]23,24] for self-complementary uniform hypergraphs. In particular, Peisert [21] gave a complete classification for symmetric (i.e.…”

Cyclic partitions of complete uniform hypergraphs

“…The vertex transitive self-complementary k-uniform hypergraphs are the subject of the paper [11] by Potǒcnik andŠajna. Gosselin gave an algorithm to construct some special self-complementary k-uniform hypergraphs in [3]. In [6] and [10] Knor, Potǒcnik andŠajna study the existence of regular self-complementary k-uniform hypergraphs.…”