What is a cycle hypergraph? What is a cycle hypergraph? Could someone give me good reference or illustrate with a few examples?
 A: This can be one visual alternative. Concepts of hypergraphs won’t be addressed in this answer. In any case, the paper “Directed hypergraph and applications” of Giorgio Gallo, Giustino Longo, Sang Nguyen,and Stefano Pallottino is a good support.
In order to explain the concept, i'll use a form of graph coloring applied to hypergraphs.
Contact Coloring and Hypergraphs
Every pair hyperedge-vertex will be called contact.

The action of coloring contacts in a hypergraph will be called “Contact Coloring of a Hypergraph”.

The contacts of a hypergraph are divided in two groups, the contacts whose vertices are in the tail of a hyperedge will be called tail contacts, and those whose vertices are in the head, will be called head contacts.
Moving through a hyperedge of a directed hypergraph is moving from a vertex that belongs to the tail of the hyperedge to a vertex that belongs to the head of the hyperedge, then it is said that the hyperedge was passed.

The action of assign a same color to all the head contacts that belongs to a directed hypergraph will be called “Contact Coloring of directed Hypergraphs”.

Also is possible to assign colors to the head contacts of a path of a certain hypergraph, that will be called “coloring the head contacts that belong to a path of a certain hypergraph”, o shortly, “coloring contact of a directed path”, at the same time, an alternative way to represent a path of a certain directed path of a hypergraph.
Properties of a Cycle Hypergraph of level N and length L


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*It has only hyperedges with N vertices, and only vertices of degree N.

*It has L number of vertices, and L number of hyperedges.

*It is possible to find at least a set of N directed paths, where:


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*Every path has a length of L.

*Every hyperedges is a backward-hyperedge.

*It is possible pass through each hyperedge once (Eulerian path).

*It is possible pass through each vertex once (Hamiltonian path).

*If N = L, we going to call it trivial cycle hypergraph, on the contrary, it will be called non-trivial cycle hypergraph.



Visualization of the features of a Cycle Hypergraph


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*In order to visualize the eulerian property of a cycle hypergraph, we are going to use the directed versions of the hypergraph.

*In order to visualize the hamiltonian property of a cycle hypergraph, we are going to use the versions of the hypergraph with directed contact coloring, all combined harmoniously in the same hypergraph.


Trivial hypergraph of level 2

(source: twimg.com) 
Trivial hypergraph of level 3

Non-trivial cycle hypergraph of level 2 and length of 5

Non-trivial cycle hypergraph of level 3 and length of 27





A: There could be more than one definition. One is $V=\{v_1,\dots,v_n\}, E = \{(v_1,v_2,v_3),(v_3,v_4,v_5),(v_5,v_6,v_7),\dots(v_{n-1},v_n,v_1)\}$.
Another one is $V=\{v_1,\dots,v_n\}, E = \{(v_1,v_2,v_3),(v_2,v_3,v_4),(v_3,v_4,v_5),\dots,(v_{n-1},v_n,v_1),(v_n,v_1,v_2)\}$.
