Abstract
A vertex irregular total -labeling of a graph with vertex set and edge set is an assignment of positive integer labels to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of , denoted by is the minimum value of the largest label over all such irregular assignment. In this paper, we consider the total vertex irregularity strengths of disjoint union of isomorphic sun graphs, , disjoint union of consecutive nonisomorphic sun graphs, , and disjoint union of any two nonisomorphic sun graphs .
1. Introduction
Let be a finite, simple, and undirected graph with vertex set and edge set . A vertex irregular total -labeling on a graph is an assignment of integer labels to both vertices and edges such that the weights calculated at vertices are distinct. The weight of a vertex in is defined as the sum of the label of and the labels of all the edges incident with , that is,
The notion of the vertex irregular total -labeling was introduced by Bača et al. [1]. The total vertex irregularity strength of , denoted by , is the minimum value of the largest label over all such irregular assignments.
The total vertex irregular strengths for various classes of graphs have been determined. For instances, Bača et al. [1] proved that if a tree with pendant vertices and no vertices of degree 2, then . Additionally, they gave a lower bound and an upper bound on total vertex irregular strength for any graph with vertices and edges, minimum degree and maximum degree , . In the same paper, they gave the total vertex irregular strengths of cycles, stars, and complete graphs, that is, , and .
Furthermore, the total vertex irregularity strength of complete bipartite graphs for some and had been found by Wijaya et al. [2], namely, for , for , for , for , and for all and . Besides, they gave the lower bound on for , that is, . Wijaya and Slamin [3] found the values of total vertex irregularity strength of wheels , fans , suns and friendship graphs by showing that , , , .
Ahmad et al. [4] had determined total vertex irregularity strength of Halin graph. Whereas the total vertex irregularity strength of trees, several types of trees and disjoint union of copies of path had been determined by Nurdin et al. [5–7]. Ahmad and Bača [8] investigated the total vertex irregularity strength of Jahangir graphs and proved that , for and conjectured that for and , They also proved that for the circulant graph, , and conjectured that for the circulant graph with degree at least 5, , .
A sun graph is defined as the graph obtained from a cycle by adding a pendant edge to every vertex in the cycle. In this paper, we determine the total vertex irregularity strength of disjoint union of the isomorphic sun graphs , disjoint union of consecutive nonisomorphic sun graphs and disjoint union of two nonisomorphic sun graphs , as described in the following section.
2. Main Results
We start this section with a lemma on the lower bound of total vertex irregularity strength of disjoint union of any sun graphs as follows.
Lemma 2.1. The total vertex irregularity strength of disjoint union of any sun graphs is , , and .
Proof. The disjoint union of the isomorphic sun graphs has vertices of degree 1 and vertices of degree 3. Note that the smallest weight of vertices of must be 2. It follows that the largest weight of vertices of degree 1 is at least and of vertices of degree 3 is at least . As a consequence, at least one vertex or one edge incident with has label at least . Moreover, at least one vertex or one edge incident with has label at least . Then Because of then
We now present a theorem on the total vertex irregularity strength of disjoint union of the isomorphic sun graphs as follows.
Theorem 2.2. The total vertex irregularity strength of the disjoint union of isomorphic sun graphs is , for and .
Proof. Using Lemma 2.1, we have . To show that , we label the vertices and edges of as a total vertex irregular labeling. Suppose the disjoint union of the isomorphic sun graphs has the set of vertices and the set of edges The labels of the edges and the vertices of are described in the following formulas: The weights of the vertices and of are It is easy to see that the weights calculated at vertices are distinct. So, the labeling is vertex irregular total. Therefore for and .
Figure 1 illustrates the total vertex irregular labeling of the disjoint union 5 copies sun graphs .
If we substitute into the theorem above, we obtain a result that has been proved by Wijaya and Slamin [3] as follows.
Corollary 2.3. The total vertex irregularity strength of sun graph , for and .
The following theorem shows the total vertex irregularity strength of disjoint union of nonisomorphic sun graphs with consecutive number of pendants.
Theorem 2.4. The total vertex irregularity strength of disjoint union of consecutive nonisomorphic sun graphs is , for .
Proof. Using Lemma 2.1, we have . To show that , we label the vertices and edges of as a total vertex irregular labeling. Suppose the disjoint union of the nonisomorphic sun graphs with consecutive number of pendants has the set of vertices and the set of edges The labels of the edges and the vertices of are described in the following formulas: The weights of the vertices and of are It is easy to see that the weights calculated at vertices are distinct. So, the labeling is vertex irregular total. Therefore for dan .
Figure 2 illustrates the vertex irregular total 10 labelings of the disjoint union 4 consecutive nonisomorphic sun graphs .
Finally, we conclude this section with a result on the total vertex irregularity strength of disjoint union of two nonisomorphic sun graphs as follows.
Theorem 2.5. The total vertex irregularity strength of disjoint union of two nonisomorphic sun graphs is , for .
Proof. Using Lemma 2.1, we have . To show that , we label the vertices and edges of as a vertex irregular total -labeling. Suppose the disjoint union of the nonisomorphic sun graphs with different pendant has the set of vertices and the set of edges The labels of the edges and the vertices of are described in the following formulas: The weights of the vertices and of are It is easy to see that the weights calculated at vertices are distinct. So, the labeling is vertex irregular total. Therefore for .
Figure 3 illustrates the vertex irregular total 6 labelings of the disjoint union of 2 nonisomorphic sun graphs .
3. Conclusion
We conclude this paper with the following conjecture for the direction of further research in this area.
Conjecture 1. The total vertex irregularity strength of disjoint union of any sun graphs is , for , and .