A note on edge irregularity strength of some graphs

Let G(V,E) be a finite simple graph and k be some positive integer. A vertex k-labeling of graph G(V,E), φ : V → {1, 2, . . . , k}, is called edge irregular k-labeling if the edge weights of any two different edges in G are distinct, where the edge weight of e = xy ∈ E(G), wφ(e), is defined as wφ(e) = φ(x) + φ(y). The edge irregularity strength for graph G is the minimum value of k such that φ is irregular edge k-labeling for G. In this note we derive the edge irregularity strength of chain graphs mK3 − path for m 6≡ 3(mod 4) and C[C n ] for all positive integers n ≡ 0(mod 4) and m. We also propose bounds for the edge irregularity strength of join graph Pm + Kn for all integers m,n ≥ 3.


Introduction
Let G(V, E) be a finite simple graph and k be some positive integer. A vertex k-labeling of graph G(V, E), φ : V → {1, 2, . . . , k}, is called edge irregular k-labeling if the edge weights of any two different edges in G are distinct, where the edge weight of e = xy ∈ E(G), w φ (e), is defined as w φ (e) = φ(x) + φ(y). The edge irregularity strength for graph G is the minimum value of k such that φ is irregular edge k-labeling for G.

(m)
n ] is frequently denoted by mK n − path.
Edge irregularity strength for some graphs have been established. (See eg. [1], [4], [5]). In this note we derive the edge irregularity strength of chain graphs mK 3 − path for m ≡ 3(mod 4) and C[C (m) n ] for all positive integers n ≡ 0(mod 4) and m. We also propose bounds for the edge irregularity strength of join graph P m + K n for all integers m, n ≥ 3.

Main Results
Through out this paper, we restrict our discussion only for finite simple graph. Let G be a graph. For some vertex v ∈ G, d(v) stands for the degree of the vertex v. The maximum degree of G, ∆(G), is defined as the maximum value of d(v), v ∈ G. The label of vertex v will be frequently denoted by l(v). Moreover, if x is a real number and s is the smallest integer such that s ≥ x, then we write s = x .
The following lemma is for facilitating the proof of Theorem 2.1. Proof. We will only prove the a part of the lemma. The b part is omitted. The proof is carried out using mathematical induction principles for r. For r = 1, by inspection we can see that the relation is true. Now assume the lemma is true for any positive integer r = s. Thus we have 3(2s+3) 2 = 5 + 3s. Consider r = s + 1. We have Based on the induction assumption, 3(2s+3) 2 = 5 + 3s, we obtain 3(2(s+1)+3) 2 = 5 + 3(s + 1). Therefore, we may conclude that 3(2r+3) 2 = 5 + 3r for all positive integer r. www.ijc.or.id A note on edge irregularity strength of some graphs | I N. Suparta and I G. P. Suharta We will start the main discussion with a fundamental theorem on edge irregularity strength of simple graphs.
Regarding the irregularity strength of mK 3 − path, we present the following theorem which is due to Ahmad, Gupta, and Simanjuntak [1].  With respect to this problem, in this discussion we derive a partial solution for it, that is for m ≡ 3(mod 4). The following Subsection 2.1 will give explanation on how we derive the partial answer.

Irregularity Strength of mK 3 − path
As a motivation we will show some examples of some labeling of mK 3 − path with their irregularity strengths: chain graph 4K 3 − path with es(4K 3 -path) = 8 = 3(m+1) 2 (see Figure 1); chain graph 5K 3 − path with es(5K 3 − path) = 9 = 3(m+1) 2 (see Figure 2).  In the following, we will show how we define an irregular labeling for mK 3 − path, m ≡ 3(mod 4), and show that es(mK 3 First, we denote the vertices of chain graph mK 3 − path as we see in Figure 3. Thus, the chain graph mK 3 − path has the following elements: • V (mK 3 -path) = {x i : 1 ≤ i ≤ m + 1} ∪ {y j : 1 ≤ j ≤ m}, and www.ijc.or.id A note on edge irregularity strength of some graphs | I N. Suparta and I G. P. Suharta We will proceed using mathematical induction principles for m. We consider two cases on m: m even and m ≡ 1(mod 4). Case m even. First we introduce two 2K 3 − paths having different irregular labeling. The one in Figure 4 we call as adder A and the other in Figure 5 we call as adder B. Remark From these two adders we have the following important observations with respect to inductive process: If we add by 3 all vertex labels of adder A (resp. adder B), then from the resulting labeling we get that l(x 3 ) of adder A(resp. adder B) is the same as l(x 1 ) of adder B(resp. adder A). Then we identify these two vertices x 3 of adder A(adder B) and x 1 of adder B(adder A), to have 4K 3 − path with an irregular labeling. This remark is indeed needed for concluding the labeling irregularity of the resulting graph through mathematical induction process. We will call a derivation graph for the resulting graph which is obtained by adding all vertex labels of graph with the same constant (in this instance the constant is 3).
Furthermore, we create a seed graph 2K 3 −path as is shown in Figure 6, for the commencement of inductive process. Here m = 2, and we can immediately see that this graph has edge irregular labeling with es(2K 3 − path) = 5 = 3(m+1) 2 . The next process of induction is conducted as follows. All labels of this seed graph 2K 3 − path are added up by constant 3. The resulting graph 2K 3 −path will have l(x 3 ) = 2+3 = 5. It is easy to see that the irregularity of induced edge labels www.ijc.or.id A note on edge irregularity strength of some graphs | I N. Suparta and I G. P. Suharta are maintained for the derivation graph, since all vertex labels increase to the same constant 3. This irregularity property always holds any time we produce derivation graphs. Then we identify vertex This label is the same as l(x 1 ) of adder B. Thus, we identify vertex x 3 of the derivation of 2K 3 −path with x 1 of adder B. The resulting chain graph 4K 3 −path has edge irregular labeling with es(4K 3 − path) = 8 = 3(4+1) 2 . This chain graph is shown in Figure 1. Now observe the derivation of the resulting 4K 3 − path. Since the rightmost two blocks of the resulting chain graph 4K 3 − path are adder B, as described in the remark, we can identify vertex x 5 from the derivation of 4K 3 − path with vertex x 1 of adder A. The resulting chain graph 6K 3 − path is shown in Figure 7.
Let m = 2l for some positive integer l. Continue this identifying process to produce (m + 2)K 3 −path from the derivation of mK 3 −path and adder A or from the derivation of mK 3 −path and adder B as follows: If l is even, we identify vertex x m+1 from the derivation of mK 3 −path with vertex x 1 of adder A, and if l is odd, we identify vertex x m+1 from the derivation of mK 3 − path with vertex x 1 of adder B. Let r be the number of times we repeat identification process for producing chain graph (2 + 2r)K 3 − path. We see that each identification process results in the increase of irregularity strength by 3. Since the seed chain graph 2K 3 − path has es(2K 3 − path) = 5, then we get that es((2 + 2r)K 3 − path) = 5 + 3r which is by Lemma 2.1, equal to 3(2r+3) . Therefore, we can conclude that for case m even, we obtain that es(mK 3 − path) = 3(m+1) 2 . Now we discuss for case m ≡ 1(mod 4). It is clear that es(K 3 − path) = 3. So, for m = 1 we have es(mK 3 − path) = 3 = 3(m+1) 2 . For m = 5, an irregular labeled chain graph 5K 3 − path is shown in Figure 2. Observe that all induced edge labels of this chain graph take all integers from 3 up until 17. This indicates that the largest vertex label is the irregularity strength of the graph. Thus, es(5K 3 − path) = 9.
We will use this chain graph as the seed graph to construct an irregular labeling for chain graph mK 3 − path with m ≥ 5. Note that l(x 6 ) = 5. Furthermore, we construct a chain graph 4K 3 − path, with irregular labeling as is shown in Figure 9. Observe that the induced edge labels of this adder C also run from 3 up to 14, and that l(x 1 ) = 11 and l(x 5 ) = 5. The technique we use to produce bigger mK 3 − path is the same as the technique we applied for m even. First add by 6 to all vertex labels of mK 3 − path, m ≥ 5. The resulting derivation graph will also have irregular labeling with the smallest edge label equals 3+6+6 = 15. This edge label is the successor of the largest edge label of adder C which is equal to l(x 1 x 2 ) = 14. Moreover, we have l(x m+1 ) = 5 + 6 = 11 in the derivation graph which is the same as l(x 1 ) in adder C. We identify these two vertices to produce (m + 4)K 3 − path having irregular labeling. For m = 5, we have es((m + 4)K 3 − path) = 9 + 6 = 15 as is shown in Figure 10. Continuing this identification process, we see that the resulting chain graph (m + 4)K 3 − path has l(x m+1 ) = l(x 10 ) = 5 because this comes from l(x 5 ) of adder C. Continuing this process we will have irregular labeling for mK 3 − path, with m ≥ 5. The irregularity strength of the chain graph mK 3 −path is derived as follows. We know already that es(5K 3 −path) = 9. Let r stand for the number of times we do identification processes. Thus, after conducting r times identification processes, we have m = 5 + 4r and es(mK 3 − path) = 9 + 6r. Using Lemma 2.1 we can conclude that es(mK 3 − path) = 9 + 6r = 3(4r+6) Therefore, for case m ≡ 1(mod 4) we also have that es(mK 3 − path) = 3(m+1) 2 . Based on this above observation we proved already the following theorem. Based on this result, with respect to irregularity strength of mK 3 −path, we have the following reduced open problem instead of the one proposed by Ahmad, Gupta, and Simanjuntak in [1].
Open Problem 2. For any positive ingeter m ≡ 3(mod 4), determine the es(mK 3 − path).  Here we also address this formulated conjecture, and introduce a solution for n ≡ 0(mod 4) as we describe below. First we name vertices of C[C (m) n ] as shown in Figure 11. Therefore, the graph C[C  . Since n is even, we have that mn+1 2 = mn 2 + 1 2 = mn 2 + 1. This is formulated as the following theorem. Proof. To this, we label graph C[C (m) n ] using the following function where N 1 := max{s ∈ Z + : s ≤ n − 3, s ≡ 1(mod 4)} and N 3 := max{s ∈ Z + : s ≤ n − 3, s ≡ 3(mod 4)}.

Irregularity strength of C[C
Note that the largest vertex label is equal to f (y 0 ) = mn 2 + 1. So, we need only to show that the labeling function f gives irregular labeling for C[C (m) n ]. This will be completed as the following.
Let us see the case m = 1. Using the above labeling function f , we can see that: So, the edge labels of C[C n ] is irregular, we may conclude that f is irregular labeling for all positive integer m and all positive integer n ≡ 0(mod 4). Thus we may conclude that the theorem is proved.
We mention again that this result confirms a partial portion of above Conjecture 1. Therefore, the remaining problem now is as the following conjecture.

Conjecture 2.
For potive integers m ≥ 2, n ≥ 5, n ≡ 0(mod 4), the edge irregularity strength of 2.3. Irregularity strength for P m + K n Figure 12. Vertex names for P m + K n Now we will proceed to address irregularity strength of join graph P m + K n which is also discussed in [1] for certain cases. Some bounds for the irregularity strength of this graph for all integers n ≥ 3 and 3 ≤ m ≤ 6 were proposed in [1].
In this paper, we derive bounds for the irregularity strength of the graph for all integers m ≥ 3 and n ≥ 3. We will see later that for 3 ≤ m ≤ 6 and n ≥ 3, our bounds are the same as in [1]. Therefore, our bounds can be considered as some extension of those in [1].
Before we formulate the bounds, we will start the process by firstly naming vertices of graph P m + K n as in Figure 12. So, the graph has elements as follows. Proof. For the lower bound follows the result which is formulated in Theorem 2.1. Here we only discuss the upper bound in the theorem. Consider again the diagram of graph P m + K n as is shown in Figure 12. Now we introduce the following labeling function for P m + K n .