Browsing by Subject "cycle decomposition"

Now showing 1 - 4 of 4

Access status: Open Access ,
Decomposing complete 3-uniform hypergraph Kn(3) into 7-cycles
(Wydawnictwa AGH, 2019) Meihua; Guan, Meiling; Jirimutu
We use the Katona-Kierstead definition of a Hamiltonian cycle in a uniform hypergraph. A decomposition of complete $k$-uniform hypergraph $K^{(k)}_{n}$ into Hamiltonian cycles was studied by Bailey-Stevens and Meszka-Rosa. For $n\equiv 2,4,5\pmod 6$, we design an algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of $K^{(3)}_{n}$ into 5-cycles has been presented for all admissible $n \leq 17$, and for all $n=4^{m}+1$ 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 $\lambda=\frac{(n-1)(n-2)}{42}$ sequences $(k_{i_{0}},k_{i_{1}},\ldots,k_{i_{6}})$ on $D_{all}(n)$, then $K^{(3)}_{n}$ can be decomposed into 7-cycles. We use the method of edge-partition and cycle sequence. We find a decomposition of $K^{(3)}_{37}$ and $K^{(3)}_{43}$ into 7-cycles.
Access status: Open Access ,
Decompositions of complete 3-uniform hypergraphs into cycles of constant prime length
(Wydawnictwa AGH, 2020) Lakshmi, Rangarajan; Poovaragavan, T.
A complete $3$-uniform hypergraph of order $n$ has vertex set $V$ with $|V|=n$ and the set of all $3$-subsets of $V$ as its edge set. A $t$-cycle in this hypergraph is $v_1, e_1, v_2, e_2,\dots, v_t, e_t, v_1$ where $v_1, v_2,\dots, v_t$ are distinct vertices and $e_1, e_2,\dots, e_t$ are distinct edges such that $v_i, v_{i+1}\in e_i$ for $i \in \{1, 2,\dots, t-1\}$ and $v_t, v_1 \in e_t$. 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 $3$-uniform hypergraph of order $n$ into $p$-cycles, whenever $p$ is prime.
Access status: Restricted ,
Systemy $k$-cykli
(Data obrony: 2012-12-17) Ginter, Aleksandra
Wydział Matematyki Stosowanej
Access status: Open Access ,
Vertices with the second neighborhood property in Eulerian digraphs
(Wydawnictwa AGH, 2019) Cary, Michael
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.