## Shonda Gosselin ; Andrzej Szymański ; Adam Pawel Wojda - Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

dmtcs:604 - Discrete Mathematics & Theoretical Computer Science, August 24, 2013, Vol. 15 no. 2 - https://doi.org/10.46298/dmtcs.604
Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

Authors: Shonda Gosselin ; Andrzej Szymański ; Adam Pawel Wojda

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.

Volume: Vol. 15 no. 2
Section: Combinatorics
Published on: August 24, 2013
Submitted on: November 26, 2010
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]