TOTAL EDGE IRREGULAR LABELING FOR TRIANGULAR GRID GRAPHS AND RELATED GRAPHS

ABSTRACT


INTRODUCTION
Let Γ = ( Γ , Γ ) be a simple, undirected, and connected graph where Γ and Γ are the set of vertices and edges of Γ, respectively. A map that assigns some set of elements of graph Γ to the set of positive or nonnegative integers is spoken as labeling. The domain of this map can only be the set of vertices (vertex labeling), the set of edges (edge labeling), or the union of vertex and edge set (total labeling) [1].
In this paper, we only discuss about particular case of total labeling i.e. total edge irregular -labeling. In mathematical word, graph Γ is a total edge irregular -labeling graph if there exists a map ∶ Γ ∪ Γ → {1,2, … , } such that for any , ∈ Γ , ( ) ≠ ( ). We called ( ) as the weight of edge and it is defined as ( ) = ( ) + ( ) + ( ). The minimum for which exists is spoken as the strength of total edge irregular labeling of Γ, represented by (Γ). Let Δ(Γ) be a maximum vertex degree of Γ. Bača, et al. [1] showed that the tes of any given graph Γ is at least In fact, all graphs are conjectured by Ivančo  Let , ∈ ℝ, so that ≤ ⌈ ⌉ and ≤ ⌈ ⌉. Then and for all ∈ ℤ.
In this paper, we will show that Equation (1) is also true for three types of graphs, namely for triangular grid graphs, some spanning subgraphs of triangular grid graph, and some Sierpiński gasket graphs.

RESULTS AND DISCUSSION
In this section, we will discuss triangular grid graph and its spanning subgraphs and Sierpiński gasket graphs, from the terminology of each graph up to the result on their total edge irregularity strength. In particular, by proving tes of triangular grid graph and its spanning subgraph, we do explain in a certain way to get the labels by seeing the structure of those graphs.

Triangular Grid Graphs
Triangular grid graph = ( , ) of levels is a graph obtained by piling up , respectively. For = 2,3, … , , let be the weight of the last diagonal edges at the th level and let ℎ be the weight of the last horizontal edges at the th level. In graph , we have = ( −1, , +1 ) and ℎ = ( , , +1 ). These terms may be important to prove the of any given graph especially triangular grid graphs and related graphs. To do that, we first determine an explicit formula of and ℎ by using ℎ (prescribed) where < , and then use and ℎ to find the weights and labels of every edge at th level. The index for and ℎ are depending on the regular pattern labels appear at the first time such that it might be distinct for every graph. The following theorem describes the total edge irregularity strength of for any ∈ ℕ.

Theorem 1. For every positive integer , it follows that
Proof. It is easy to check that for all ∈ ℕ, Δ( ) ≤ ⌈ 3 ( +1) +4 6 ⌉. Therefore, we obtain that ( ) ≥ For = 2, we prescribe a total edge irregular 4-labeling for 2 by Figure 2. We know that ℎ 2 = 11, so that for = 3 we have the weights of diagonal and horizontal edges provided in Table 1.
From Table 1, we obtain 3 = ℎ 2 + 6 and ℎ 3 = ℎ 2 + 9. If we continue this observation for ≥ 4, we will obtain = Clearly all weights are distinct and we realized that the last diagonal and horizontal edge label of th level ( = 3,4, … , ) are always less than ⌈ 3 ( +1) +4 6 ⌉ because of the following results.
a. Last horizontal edge label Hence, the proof is completed. ∎ Figure 3 illustrates a graph 4 with a total edge irregular 11-labeling.

Spanning Subgraphs of Triangular Grid Graph
Now we come to the first spanning subgraph of triangular grid graph. This graph is a triangular grid graph without two border edges of each level, denoted by 2 for all positive integer (see Figure 4).

Figure 4. Graph
In this part, the Theorem 2 and Theorem 3 will be proved by using the terms and ℎ such like the previous subsection.

Theorem 2. For every positive integer , it follows that
Proof. It is easy to check that for any ∈ ℕ, Δ( 2 ) ≤ ⌈

Figure 5. Graph with total edge irregular -labeling TOTAL EDGE IRREGULAR LABELING FOR TRIANGULAR GRID GRAPHS AND…
We know that ℎ 2 = 9. By doing some observations for such ≥ 4, we will obtain = 3 ( −1)+8 2 and ℎ = The Inequality (5) contradicts the fact that the ceiling of any real number is greater than or equal to . Therefore, we obtain ( , , +1 ) ≤ ⌈ ⌉.
Hence, the proof is completed. ∎ Figure 6 illustrates a graph 2 5 with a total edge irregular 13-labeling.

Figure 6. Graph with a total edge irregular -labeling
The second spanning subgraph of triangular grid graph that we observe is also with some modifications. We remove one border edge of each level such that there does not exist a pair of two incidence border edges which are removed together. This graph is denoted by 1 . It is clear that | 1 | = | | and for ∈ ℕ (see Figure 7). Obviously, the vertex set is 1    We know that ℎ 4 = 28. By doing some observations for ≥ 6, we will obtain and ℎ as follows The Inequality (6) contradicts the fact that the ceiling of any real number is greater than or equal to . Therefore, we obtain ( , , +1 ) ≤ ⌈ (3 +1)+4 6 ⌉ where = 5,6, … , .
• Last diagonal edge label For is even, ⌉ for is odd.
Hence, the proof is completed. ∎ The tes of the mirror of 1 is clearly equivalent to the Theorem 3. The vertex labels are similar with 1 but the edge labels are different with 1 , precisely it is different at index and . Explicitly, we obtain horizontal edge label is

Sierpiński Gasket Graphs
Sierpiński gasket is a geometric shape formed by infinitely repeated dividing a triangle into four smaller triangles out of its center, whereas Sierpiński gasket graph ( ) of levels is a graph obtained by − 1 repeated dividing a triangle graph into smaller triangle graphs out of its center. In other words, consists of three attached copies of −1 which refer to as top, bottom left, and bottom right components of , denoted by , , , , and , , respectively (see Figure 11). It is easy to see that Sierpiński gasket graph is also a subgraph of triangular grid graph and it has 3 2 (3 −1 + 1) vertices and 3 edges for every positive integer ≥ 2. The following theorem as a result of our observation about the total edge irregularity strength for some cases of Sierpiński gasket graphs. i. For = 1, is isomorphic with triangular grid graph 1 and cycle 3 . By Figure 5 and [1], we obtain ( 1 ) = 2.