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Decompositions of complete 3-uniform hypergraphs into cycles of constant prime length

Lakshmi R, Poovaragavan T.

Opuscula Mathematica

2020

Vol. 40, no. 4

509--516

A complete 3-unilorm hypergraph of order n has vertex set V with \V\ = n and the set ol all 3-subsets of V as its edge set. A t-cycle in this hypergraph is v1, e1, v2, e2,… , vt, et, v1 where v1, v2,…vt are distinct vertices and e1, e-2,..., et are distinct edges such that [formula] and [formula] A decomposition of a hypergraph is a partition of its edge set into edge-disjoint subsets. In this paper, we give necessary and sufficient conditions for a decomposition of the complete 3-unilorm hypergraph of order n into p-cycles, whenever p is prime.

Vertices with the second neighborhood property in Eulerian digraphs

Cary Michael

Opuscula Mathematica

2019

Vol. 39, no. 6

765--772

The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph not only adhere to the Second Neighborhood Conjecture, but that local simplicity can, in some cases, also imply the existence of a Seymour vertex in the original digraph.

Decomposing complete 3-uniform hypergraph kn(3) into 7-cycles

Meihua, Guan Meiling, Jirimutu

Opuscula Mathematica

2019

Vol. 39, no. 3

383-–393

We use the Katona-Kierstead definition of a Hamiltonian cycle in a uniform hypergraph. A decomposition of complete k-uniform hypergraph [formula] into Hamiltonian cycles was studied by Bailey-Stevens and Meszka-Rosa. For n ≡ 2,4, 5 (mod 6), we design an algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of [formula] into 5-cycles has been presented for all admissible n ≤ 17, and for all n = 4m + 1 when m is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we show if 42 | (n — 1)(n — 2) and if there exist [formula] sequences (ki0, ki1,…..,k16) on Dall(n), then [formula] can be decomposed into 7-cycles. We use the method of edge-partition and cycle sequence. We find a decomposition of [formula] and [formula] into 7-cycles.