Total vertex irregularity strength for trees with many vertices of degree two

For a simple graph G = (V,E), a mapping φ : V ∪ E → {1, 2, . . . , k} is defined as a vertex irregular total k-labeling of G if for every two different vertices x and y, wt(x) 6= wt(y), where wt(x) = φ(x)+ ∑ xy∈E(G) φ(xy). The minimum k for which the graphG has a vertex irregular total klabeling is called the total vertex irregularity strength of G. In this paper, we provide three possible values of total vertex irregularity strength for trees with many vertices of degree two. For each of the possible values, sufficient conditions for trees with corresponding total vertex irregularity strength are presented.


Introduction
The concept of total vertex irregularity strength of graphs was first introduced by Baca et.al [2] in 2007. They defined a mapping φ : V ∪ E → {1, 2, 3, . . . , k} to be a vertex irregular total k-labeling of G if for every two different vertices x and y, wt(x) = wt(y), where wt(x) = φ(x) + xy∈E(G) φ(xy). The minimum k for which the graph G has a vertex irregular www.ejgta.org Total vertex irregularity strength for trees with many vertices of degree two | R. Simanjuntak, et al.
total k-labeling is called the total vertex irregularity strength of G, denoted by tvs(G). Baca et.al determined the total vertex irregularity strength of some well-known classes of graphs, i.e. paths, cycles, and stars. Other authors (for instance, [1], [3]) determined the total vertex irregularity strength of some other classes of graphs, however results are still limited.
In the original paper of Baca et.al [2], it was proved that for a tree T with m pendant vertices and no vertex of degree 2, The lower bound in Theorem 1.1 remains the most general bound known for trees. However, it was conjectured that the total vertex irregularity strength of a tree is only determined by the number of vertices of degrees at most 3. To date, the conjecture has been confirmed for some types of trees, i.e. paths and stars, trees with maximum degree up to 5 [4,6,7] and subdivision of some classes of trees [5,8].
In this paper, our aim is to determine the total vertex irregularity strength of trees with many vertices of degree 2 which include subdivision of trees. This result could somewhat be viewed as generalization of our result in [8], where we presented sufficient conditions for subdivision of trees to admit total vertex irregularity strength of t 2 .
Throughout the paper, we consider T as a tree with maximum degree ∆. We denote by n i the number of vertices of degree i(i = 1, 2, . . . , ∆) and t i =

Basic Properties of Trees
In this section, we shall provide properties of trees, in term on n 1 , n 2 , and n 3 , having t 1 , t 2 or t 3 as the maximum among all t i s. We start by quoting a useful property proved in [2].

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Total vertex irregularity strength for trees with many vertices of degree two | R. Simanjuntak, et al.

Trees with Many Vertices of Degree 2
In this section, we provide sufficient conditions, in term on n 1 , n 2 , and n 3 , for a tree T with many vertices of degree 2 admitting tvs(T ) = t 1 , t 2 or t 3 .
We start by defining several notions that will be frequently utilized in our labeling algorithms. Let v be a vertex of T . A branch of T at v is defined as maximal subtree of T containing v as an end point. That is, a branch of T at v is the subgraph induced by v and one of the components of T − v. If the degree of v is k, then v has k different branches. A branch of T at v which isomorphic to a path will be called a branch path at v, provided that the degree of v is at least 3. The vertex v, in this case, will be called a stem of the branch path at v. We define an interior path in T as a path whose both of end vertices are stem vertices. A vertex of degree one in T is called a pendant vertex. A vertex incident to a pendant vertex in T is called an exterior vertex. The vertices other than exterior and pendant vertices are called interior vertices. An edge incident with a pendant vertex is called a pendant edge. We denote by E p (v) the set of pendant edges incident to an exterior vertex v.

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Total vertex irregularity strength for trees with many vertices of degree two | R. Simanjuntak, et al.
Examples of families of trees admitting total vertex irregularity strength of t 2 are special cases of subdivision of tress that could be found in [8].

Conclusion
Our results provide sufficient conditions for trees containing many vertices of degree 2 where the total vertex irregularity strength is either t 1 , t 2 or t 3 . These results strengthens the conjecture Nurdin et.al.