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Edge‐transitive homogeneous factorizations of complete uniform hypergraphs
Abstract:For a finite set V and a positive integer k with urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0001, letting urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0002 be the set of all k‐subsets of V, the pair urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0003 is called the complete k‐hypergraph on V, while each k‐subset of V is called an edge. A factorization of the complete k‐hypergraph urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0004 of index urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0005, simply a urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0006‐factorization of order n, is a partition urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0007 of the edges into s disjoint subsets such that each k‐hypergraph urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0008, called a factor, is a spanning subhypergraph of urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0009. Such a factorization is homogeneous if there exist two transitive subgroups G and M of the symmetric group of degree n such that G induces a transitive action on the set urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0010 and M lies in the kernel of this action. In this article, we give a classification of homogeneous factorizations of urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0011 that admit a group acting transitively on the edges of urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0012. It is shown that, for urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0013 and urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0014, there exists an edge‐transitive homogeneous urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0015‐factorization of order n if and only if urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0016 is one of (32, 3, 5), (32, 3, 31), (33, 4, 5), urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0017, and urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0018, where urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0019 and q is a prime power with urn:x-wiley:03649024:media:jgt22158:jgt22158-math-0020.
Keywords:edge‐transitive  homogeneous factorization  homogeneous permutation group  self‐complementary hypergraph  uniform hypergraph
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