Abstract
Combinatorics A \em cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form \F,F^θ,F^θ², \ldots, F^θ^q-1\ for some permutation θ of the vertex set V. Let Vₙ = \ 1,2,\ldots,n\. For a positive integer k, Vₙ\choose k denotes the set of all k-subsets of Vₙ. For a nonempty subset K of V_n-1, we let \mathcalKₙ^(K) denote the hypergraph ≤ft(Vₙ, \bigcup_k∈ K Vₙ\choose k\right). In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of \mathcalKₙ^(V_k). In particular, we prove that if p is prime then there is a cyclic p^α-partition of \mathcalK^(Vₖ)ₙ if and only if p^α + β divides n, where β = \lfloor \logₚ k\rfloor. As an application of this result, we obtain two sufficient conditions on n₁,n₂,\ldots,n_t, k, α and a prime p for the existence of a cyclic p^α-partition of the complete t-partite k-uniform hypergraph \mathcal K^(k)_n₁,n₂,\ldots,n_t.
Highlights
For a finite set V and a positive integer k, letV k denote the set of all k-subsets of V
A hypergraph is a pair (V, E) in which V is a finite set of vertices and E is a collection of subsets of V called edges
The order of a hypergraph is the cardinality of its vertex set, and the rank of a hypergraph is the maximum cardinality of an edge
Summary
V k denote the set of all k-subsets of V. A hypergraph is a pair (V, E) in which V is a finite set of vertices and E is a collection of subsets of V called edges. . The order of a hypergraph is the cardinality of its vertex set, and the rank of a hypergraph is the maximum cardinality of an edge. The vertex set and the edge set of a hypergraph X will often be denoted by V (X) and E(X), respectively. N} and we always assume that the vertex set of a hypergraph of order n is equal to Vn. The complete k-uniform hypergraph of order n is. For a nonempty subset K of Vn−1, the complete K-hypergraph is Vn, and k∈K denoted by Kn(K). Q − 1} with edge ranks in K which decompose Kn(K), and which are permuted cyclically under the action of θ. Each of the hypergraphs Xi in this decomposition are sometimes called q-complementary, and θ is called a q-antimorphism of Xi
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