Cycle in $3-$Uniform Hypergraph Could someone, please, define a cycle in a  $3-$Uniform Hypergraph? I looked at different resources, but they have different definitions and I am so confused!!
 A: The problem is that there are different notions of what cycles are in hypergraphs. You always have to specify which you mean.
In a $3$-uniform hypergraph:


*

*A loose cycle consists of edges $\{v_1,v_2,v_3\}, \{v_3,v_4,v_5\}, \dots, \{v_{2k-1}, v_{2k}, v_1\}$. From each edge of the cycle, we pick a "first vertex" and a "last vertex" (this information is not built into the hypergraph). The last vertex of each edge is the first vertex of the next, the last vertex of the last edge is the first vertex of the first edge, and no other vertices are shared.

*A tight cycle consists of edges $\{v_1,v_2,v_3\}, \{v_2,v_3,v_4\}, \dots, \{v_{k-1},v_{k},v_1\}, \{v_k, v_1,v_2\}$. There is a sequence of vertices $v_1, v_2, \dots, v_k$, and every three consecutive vertices form an edge, as do $\{v_{k-1}, v_k, v_1\}$ and $\{v_k, v_1, v_2\}$.

*A Berge cycle is a sequence of distinct vertices and edges $v_1, e_{12}, v_2, e_{23}, v_3, \dots, v_k, e_{k1}, v_1$ such that the edge $e_{i,i+1}$ contains $v_i$ and $v_{i+1}$ (and $e_{k1}$ contains $v_k$ and $v_1$). This is different from a loose cycle because we don't ask for other vertices not to be shared between the edges.


For $r$-uniform hypergraphs when $r>3$, all these notions also exist, but we can interpolate between "loose" and "tight". We could ask, for instance, that adjacent edges share two out of $r$ vertices.
A: An example of a ("pentagon") $5-$cycle in a $3-$ uniform hypergraph.

The definition of a cycle is very similar to that for a graph it is a sequence of verticies and hyperedges such that 
\begin{eqnarray*}
a_1 E_{1,2} a_2 E_{2,3} a_3 \cdots a_k E_{k,1} a_1
\end{eqnarray*}
such that $a_i, a_{i+1} \in E_{i,i+1}$.
