# In several classes of graphs are collected whether they

In  [3], several 3X FLAG of graphs are collected whether they are SEM or not. For examples, path graphs and odd cycles are SEM, but even cycles are not. However, not much has been done concerning an SEM labeling in hypergraphs. Hence, we generalize the notion of the SEM for graph to the SEM in hypergraph.
In this paper, we will consider two classes of hypergraphs, namely, -node-uniform hyperpaths, and -node-uniform hypercycles, . These classes are the generalizations of paths and cycles in graph theory. By defining SEM labelings on “small size” hypergraphs of these two classes, we then can have algorithms to construct the SEM labelings on “bigger” hypergraphs. Finally, we can conclude that are SEM and under some conditions on and , are SEM. We note that these conditions agree with the condition for a cycle to be SEM in graphs.

Preliminaries
Let us begin with the definitions of hypergraph, and .

If there is no ambiguity, we may denote as and as . For more convenience, we let and . Notice that, by Definition 2.1, can be empty. However, in this paper, our hypergraphs consist of at least one vertex. Moreover, if for all , then is called a -uniform hypergraph and it is denoted by . We can see that is an ordinary graph.

According to Fig. 2, we can regard each hyperedge of as a combination of 3 parts. Two of them are called nodes which both consist of equal vertices and . The third part is called the middle which consist . The node part is usually the intersection part of two adjacent hyperedges except for and which only their right node and left node are the intersection parts, respectively.
Notice that, has hyperedges. Each hyperedge consists of exactly vertices and has two nodes containing exactly vertices. Thus, has totally vertices.

According to Fig. 4, each hyperedge of is similar to the hyperedge of . It is a combination of 2 nodes, and , and the middle . However, each node of is the intersection of two adjacent hyperedges. Thus, it is easy to see that has vertices. Notice that and are the path and cycle in the ordinary graph sense.
We extend the notion of the SEM labeling for a hypergraph stated as Definition 2.4.

In the case of an empty edge, , we let the sum of all vertex-labels, which is none, to be zero. If a hypergraph is a 2-uniform hypergraph, then Definition 2.4 agrees with the definition of SEM labeling in graphs (see Figs. 2, 4 and 6–13).
By Definition 2.4, for every SEM labeling of , since , we obtain which is the set of consecutive integers. Moreover, since is a constant for every , must also be the set of consecutive integers. From this observation, we state the equivalent form of the SEM labeling for hypergraph as the following theorem.

We point out here that Theorem 2.5 is a generalization of the following result found in  [5] for graphs.
By Theorem 2.5, to give an SEM labeling for a hypergraph , then it is enough to assign only the labels to vertices of the hypergraph in such a way that the assignment satisfies the necessary part of the theorem.

First, we will give the SEM labelings for and . Then, we show how to extend those SEM labelings to the SEM labeling for .

For Theorems 3.2, 3.3 and 3.4, we will give the SEM labelings for and , respectively. However, the proofs of these theorems are similar to the proof of Theorem 3.1. Thus, we omit their proofs.

Now, we give the idea used in this paper on how to extend those SEM labelings in Theorems 3.1–3.4 to the SEM labelings of . First, observe that every hypergraph has nodes. If we add two vertices to each node of , then receives another vertices and becomes .
This addition of vertices can preserve the SEM property. To see that let having vertices be SEM. Then, we add new vertices to construct . We assign the label to these vertices. Next, we make pairs from these vertices such that the sum of vertex-labels of each pair is , i.e., the vertex whose label is must be paired with the vertex whose label is for every . Now, we put these pairs of vertices to each nodes, in any order. Thus, we obtain the new labeling for . Since each hyperedge of has two nodes, it is easy to see that the sum of vertex-labels in each hyperedge increased from the old one of by . Therefore, the set of the sum of vertex-labels in each hyperedge of is the set of consecutive integers. Consequently, by Theorem 2.5, is SEM. Moreover, since has more vertices from and the sum of vertex-labels is also increased by , the magic constant for is increased by from the magic constant for . We conclude this observation as Lemma 3.5.