Totally irregular total labeling of some caterpil- lar graphs

Assume that G(V,E) is a graph with V and E as its vertex and edge sets, respectively. We have G is simple, connected, and undirected. Given a function λ from a union of V and E into a set of k-integers from 1 until k. We call the function λ as a totally irregular total k-labeling if the set of weights of vertices and edges consists of different numbers. For any u ∈ V , we have a weight wt(u) = λ(u) + ∑ uy∈E λ(uy). Also, it is defined a weight wt(e) = λ(u) + λ(uv) + λ(v) for each e = uv ∈ E. A minimum k used in k-total labeling λ is named as a total irregularity strength of G, symbolized by ts(G). We discuss results on ts of some caterpillar graphs in this paper. The results are ts(Sp,2,2,q) = ⌈ p+q−1 2 ⌉ for p, q greater than or equal to 3, while ts(Sp,2,2,2,p) = ⌈ 2p−1 2 ⌉ , p ≥ 4.


Introduction
Graph theory is one of branch of mathematics. In this field, many real life problems can be solved, especially on optimization problem [8]. Given a graph G(V, E) which is assumed as connected, simple, and undirected graph. A function that assigns a set of elements (vertex/edge) of G into a set of integers is mentioned as labeling (Wallis [12]). The labeling is said to be a total labeling if the domain is a union of vertex and edge sets.
A function f : V ∪ E → {1, 2, . . . , k} is named a vertex irregular total k-labeling if wt f (u) = wt f (v) for each u = v ∈ V (G), where wt(u) = f (u) + uz∈E f (uz) [1]. A minimum k in which there exists a vertex irregular total k-labeling of G is named as a total vertex irregularity strength (tvs) of G. Indriati et al. [4] obtained tvs of generalized helm. Recently, the tvs of comb product of two cycles and two stars has been found in [10]. Meanwhile, Nurdin et al. [11] proved tvs of tree T which does not have vertex of degree two and has n pendant nodes, i.e.

tvs(T
Further, a total k-labeling g that assigns a union of V and E into {1, 2, . . . , k} is called an edge irregular when the requirement wt(xy) = wt(x y ) is satisfied for each pair xy = x y in E(G) with wt(xy) = g(x) + g(xy) + g(y). Bača et al. [1] mentioned the minimum k required in labeling g as a total edge irregularity strength (tes) of G. The exact value of tes of generalized web graphs was given in [2]. Recent research has found tes of some n-uniform cactus chain graphs and related chain graphs [6]. In addition, tes of any tree has been given in [7], i.e. tes(T ) is equal to Furthermore, the total k-labeling g becomes a totally irregular total k-labeling if the set of all weights of vertices and edges contains distinct numbers [9]. A minimum k needed in the labeling g is named as total irregularity strength (ts) of G. Marzuki, et al. observed Different with tes and tvs, the value of ts of tree has not been obtained. In order to find ts of tree, we have started the investigation for double stars S p,q and related graphs S p,2,q ([3], [5]). In this research, we verify ts of caterpillar graphs S p,2,2,q and S p,2,2,2,p .
We use the notion of caterpillar S p,2,2,q . It is a graph which is formed from double-star S p,q by putting two vertices on the path which are connected to the two centers of stars in S p,q . The value of tes of graph S p,2,2,q can be found by (2), that is This graph has two vertices of degree two. Therefore, (1) cannot be used for determining tvs of this graph. The next theorem gives this parameter.
Proof. Without loss of generality, we can assume that p ≤ q. We know that S p,2,2,q contains p + q − 2 pendants, two vertices with degree two, one vertex with degree p, and one vertex of degree q. The smallest weight of each vertex is at least two. Each pendant vertex has the smallest weight which is not less than p + q − 1, i.e. the weight is a sum of two labels. Then, the largest number to label pendant vertices is not less than p+q−1
We observe that each vertex and each edge has been labeled with a number which is at most k = p+q−1
x wt(x) Case for p, q v s r r + 1, It is shown above, each vertex has a distinct weight under total labeling f . Therefore, tvs(S p,2,2,q ) = k = p+q−1 2 .
Furthermore, an exact value of ts of S p,2,2,q is proved in the next theorem.

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Proof. According to (3), by using Equality (4) and Theorem 1.1, the lower bound is as follows: Furthermore, we use total k-labeling λ constructed in Theorem 1.1 to get a totally irregular total k-labeling. Under labeling λ, we obtain the edge-weights below.
It can be seen that each edge has a different weight. This concludes that λ is totally irregular total k-labeling. Thus, ts(S p,2,2,q ) = k = p+q−1 2 .