On irregularity strength of diamond network

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On irregularity strength of diamond network Nurdin Hinding a , Dian Firmayasari a , Hasmawati Basir a , Martin Baˇca b , Andrea Semaniˇcová-Feˇnovˇcíková b , ∗ a Mathematics Department, Faculty of Mathematics and Natural Sciences, Hasanuddin University, Jl. Perintis Kemerdekaan Km. 10 Tamalanrea,

Makassar, Indonesia b Department of Applied Mathematics and Informatics, Technical University, Košice, Slovakia

Received 21 August 2017; received in revised form 9 October 2017; accepted 10 October 2017 Available online xxxx

Abstract In this paper we investigate the total edge irregularity strength tes(G) and the total vertex irregularity strength tvs(G) of diamond graphs Brn and prove that tes(Brn ) = ⌈(5n − 3)/3⌉, while tvs(Brn ) = ⌈(n + 1)/3⌉. c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Total edge irregularity strength; Total vertex irregularity strength; Diamond graphs

1. Introduction Let us consider a simple undirected graph G = (V (G), E(G)) without loops and isolated vertices with the vertex set V (G) and the edge set E(G). For notations not defined here, we refer the reader to [1]. A labeling of a graph is a map that carries graph elements to the numbers, usually to the positive or non-negative integers. If the domain is the vertex set or the edge set, the labelings are called respectively vertex labelings or edge labelings. If the domain is V (G) ∪ E(G) then we call the labeling a total labeling. The most complete recent survey of graph labelings is [2]. For a given edge labeling λ : E(G) → {1, 2, . . . , k}, k is a positive integer, the associated weight of a vertex ∑ x ∈ V (G) is wλ (x) = x y∈E(G) λ(x y), where the sum is over all vertices y adjacent to x. An edge labeling λ is called irregular if wλ (x) ̸= wλ (y) for every pair x, y of vertices of G. The smallest integer k for which an irregular labeling of G exists is known as the irregularity strength of G and is denoted by s(G). The notion of the irregularity strength was introduced by Chartrand et al. in [3]. For more information see [4–6] and [7]. Motivated by the irregularity strength of a graph G and an excellent book by Wallis [8], Baˇca, Jendrol’, Miller and Ryan in [9] introduced two new parameters, namely a total edge irregularity strength and a total vertex irregularity strength of a graph. A total labeling ϕ : V (G) ∪ E(G) → {1, 2, . . . , k} is called an edge irregular total k-labeling Peer review under responsibility of Kalasalingam University.

∗ Corresponding author.

E-mail addresses: [email protected] (N. Hinding), [email protected] (D. Firmayasari), [email protected] (H. Basir), [email protected] (M. Baˇca), [email protected] (A. Semaniˇcov´a-Feˇnovˇc´ıkov´a). https://doi.org/10.1016/j.akcej.2017.10.003 c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: N. Hinding, et al., On irregularity strength of diamond network, AKCE International Journal of Graphs and Combinatorics (2017), https://doi.org/10.1016/j.akcej.2017.10.003.

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Fig. 1. The diamond graph Br5 .

of the graph G if every two distinct edges x y and x ′ y ′ of G satisfy wtϕ (x y) = ϕ(x) + ϕ(x y) + ϕ(y) ̸= wtϕ (x ′ y ′ ) = ϕ(x ′ ) + ϕ(x ′ y ′ ) + ϕ(y ′ ). The total edge irregularity strength, tes(G), is defined as the minimum k for which G has an edge irregular total k-labeling. In [9] is given a lower bound on the total edge irregularity strength of a graph as follows {⌈ ⌉ ⌈ ⌉} ∆(G)+1 tes(G) ≥ max |E(G)|+2 , . (1) 3 2 The further information can be found in [10–20]. The total labeling ϕ : V (G) ∪ E(G)∑→ {1, 2, . . . , k} is called a vertex irregular total k-labeling of the graph G if the vertex weights wtϕ (x) = ϕ(x) + x y∈E(G) ϕ(x y) are different for distinct vertices, that is, wtϕ (x) ̸= wtϕ (y) for all distinct vertices x, y ∈ V (G). The total vertex irregularity strength of G, denoted by tvs(G), is the minimum k for which G has a vertex irregular total k-labeling. It is easy to see that for graphs with no component of order at most two is tvs(G) ≤ s(G). In [9] a lower and an upper bound of the total vertex irregularity strength of a graph with minimum degree δ(G) and maximum degree ∆(G) are given as follows ⌉ ⌈ p+δ(G) ≤ tvs(G) ≤ p + ∆(G) − 2δ(G) + 1. (2) ∆(G)+1 In a sense, a better lower bound gives the next theorem. Theorem 1 ([21]). Let G be a connected graph having n i vertices of degree i, i = δ(G), δ(G) + 1, . . . , ∆(G), where δ(G) and ∆(G) are the minimum and the maximum degree of G, respectively. Then ⌈ {⌈ ⌉} ∑∆(G) ⌉ ⌈ ⌉ δ(G)+ i=δ(G) n i δ(G)+n δ(G) δ(G)+n δ(G) +n δ(G)+1 , , . . ., tvs(G) ≥ max . (3) δ(G)+1 δ(G)+2 ∆(G)+1 Moreover in [21] is posed the conjecture that the equality in (3) holds. This conjecture has been verified for several classes of graphs [22–24]. Let L n ∼ = Pn □P2 , n ≥ 2, be a ladder with the vertex set V (L n ) = {vi , u i : i = 1, 2, . . . , n} and the edge set E(L n ) = {vi vi+1 , u i u i+1 : i = 1, 2, . . . , n − 1} ∪ {vi u i : i = 1, 2, . . . , n}, see [2]. If we add the edges u i vi+1 , i = 1, 2, . . . , n − 1, to the ladder L n and remove the vertex u n with both incident edges u n−1 u n and u n vn , then we obtain a triangular ladder T L n . A diamond graph Brn , n ≥ 3, is obtained by joining a single vertex x to all vertices vi , i = 1, 2, . . . , n, of a triangular ladder T L n , see [25]. The vertex set of Brn is V (Brn ) = {x} ∪ {u i : i = 1, 2, . . . , n − 1} ∪ {vi : i = 1, 2, . . . , n} and the edge set is E(Brn ) = {u i u i+1 : i = 1, 2, . . . , n − 2} ∪ {vi vi+1 : i = 1, 2, . . . , n − 1} ∪ {u i vi : i = 1, 2, . . . , n − 1} ∪ {u i vi+1 : i = 1, 2, . . . , n − 1} ∪ {xvi : i = 1, 2, . . . , n}. Thus |V (Brn )| = 2n and |E(Brn )| = 5n − 5. Fig. 1 illustrates the diamond graph Br5 . In this paper, we deal with the edge irregular total labeling and the vertex irregular total labeling of diamond graphs and determine the exact value of the total edge irregularity strength and the total vertex irregularity strength for this family of graphs. 2. Total edge irregularity strength of diamond graphs The next theorem gives the exact value of the total edge irregularity strength of the diamond graphs. Please cite this article in press as: N. Hinding, et al., On irregularity strength of diamond network, AKCE International Journal of Graphs and Combinatorics (2017), https://doi.org/10.1016/j.akcej.2017.10.003.

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Theorem 2. Let Brn , n ≥ 3, be the diamond graph. Then ⌈ ⌉ tes(Brn ) = 5n−3 . 3 Proof. Since |E(Brn )| = 5n − 5 and ∆(Brn ) = 5 for n = 3, 4 and ∆(Brn ) = n for n ≥ 5 Inequality (1) implies that {⌈ ⌉ ⌈ ⌉} tes(Brn ) ≥ max |E(Br3n )|+2 , ∆(Br2n )+1 { {⌈ ⌉ } ⌈ 5n−3 ⌉ max 5n−3 ,3 = for n = 3, 4, 3 {⌈ 5n−3 ⌉ ⌈ n+1 ⌉} 3 ⌈ 5n−3 ⌉ = max , 2 = for n ≥ 5. 3 3 Hence tes(Brn ) ≥ ⌈(5n − 3)/3⌉ for every n ≥ 3. Put k = ⌈(5n − 3)/3⌉. To show that k is an upper bound for the total edge irregularity strength of the diamond graphs Brn , we describe a total k-labeling ϕ as follows. ϕ(vi ) = i ⌈ ⌉ ϕ(u i ) = 5n−3 3

for i = 1, 2, . . . , n, for i = 1, 2, . . . , n − 1,

ϕ(x) = 1, ϕ(xvi ) = 1 ϕ(vi vi+1 ) = n + 1 − i ⌈ ⌉ ϕ(u i vi ) = n+4 3 ⌊ ⌋ ϕ(u i vi+1 ) = 4n 3 ⌊ ⌋ ϕ(u i u i+1 ) = n + 2 n3 − i

for i = 1, 2, . . . , n, for i = 1, 2, . . . , n − 1, for i = 1, 2, . . . , n − 1, for i = 1, 2, . . . , n − 1, for i = 1, 2, . . . , n − 2.

It is not difficult to see that all vertex and edge labels are at most k. The weights of edges of Brn under the total labeling ϕ are as follows: wtϕ (xvi ) = 1 + 1 + i = 2 + i,

for i = 1, 2, . . . , n,

wtϕ (vi vi+1 ) = i + (n + 1 − i) + (i + 1) = n + 2 + i, for i = 1, 2, . . . , n − 1, ⌈ ⌉ ⌈ n+4 ⌉ wtϕ (u i vi ) = 5n−3 + + i = 2n + 1 + i, for i = 1, 2, . . . , n − 1, 3 ⌈ 5n−3 ⌉ ⌊ 4n3 ⌋ wtϕ (u i vi+1 ) = + + (i + 1) = 3n + i, for i = 1, 2, . . . , n − 1, 3 ⌈ 5n−3 ⌉ ( 3 ⌊n⌋ ) ⌈ 5n−3 ⌉ wtϕ (u i u i+1 ) = + n+2 3 −i + 3 = 5n − 2 − i, 3 for i = 1, 2, . . . , n − 2. It is evident that the weights of edges successively attain consecutive values 3, 4, . . . , 5n − 3. Thus, the labeling ϕ is an optimal edge irregular total k-labeling and we have arrived at the desired result. □

3. Total vertex irregularity strength of diamond graphs In this section, we determine the exact value of the total vertex irregularity strength of the diamond graphs. Theorem 3. Let Brn , n ≥ 3, be the diamond graph. Then { 3 for n = 5, tvs(Brn ) = ⌈ n+1 ⌉ otherwise. 3 Proof. As the diamond graph Brn has 2n vertices, δ(Brn ) = 3 and ∆(Brn ) = 5 for n = 3, 4 and ∆(Brn ) = n for n ≥ 5 according to Inequality (2), we have {⌈ 2n+3 ⌉ ⌈ ⌉ =2 for n = 3, 4, 5+1 |V (Brn )|+δ(Brn ) ⌉ tvs(Brn ) ≥ = ⌈ 2n+3 ∆(Brn )+1 =3 for n ≥ 5. n+1 For n = 3, 4, 5 the optimal vertex irregular total tvs(Brn )-labelings are illustrated in Figs. 2 and 3, respectively. Please cite this article in press as: N. Hinding, et al., On irregularity strength of diamond network, AKCE International Journal of Graphs and Combinatorics (2017), https://doi.org/10.1016/j.akcej.2017.10.003.

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Fig. 2. A vertex irregular total 2-labeling of Br3 and Br4 .

Fig. 3. A vertex irregular total 3-labeling of Br5 .

For n ≥ 6 the graph Brn contains four vertices of degree 3, n − 3 vertices of degree 4, n − 2 vertices of degree 5 and one vertex of degree n. According to Inequality (3) we get {⌈ ⌉ ⌈ ⌉ ⌈ 2n+2 ⌉ ⌈ 2n+3 ⌉} ⌈ n+1 ⌉ tvs(Brn ) ≥ max 47 , n+4 , 6 , n+1 = 3 ≥ 3. (4) 5 Suppose k = ⌈(n + 1)/3⌉. To obtain the converse inequality it is sufficient to prove the existence of a vertex irregular total k-labeling for the diamond graph Brn . Let us consider three cases. Case 1. If n ≡ 0 (mod 3), n ≥ 6 we construct the total labeling ψ : V (Brn ) ∪ E(Brn ) → {1, 2, . . . , k} in the following way ψ(x) = 2, ⎧ for i = 1, n, ⎪ ⎨1 n for 2 ≤ i ≤ n3 + 2, ψ(vi ) = 3 ⎪ ⎩ k for n3 + 3 ≤ i ≤ n − 1, ⎧ k for i = 2, ⎪ ⎪ ⎪ ⎨2 for i odd, 1 ≤ i < 2n and for i = n, 3 ψ(xvi ) = 2n ⎪3 for i even, 4 ≤ i ≤ 3 , ⎪ ⎪ ⎩ 2n−3 i− 3 for 2n + 1 ≤ i ≤ n − 1, 3 ⎧ for 1 ≤ i ≤ 2n , ⎨1 3 2n for 2n + 1 ≤ i ≤ n − 2 when n ≥ 9, ψ(u i ) = i − 3 3 ⎩ 2 for i = n − 1, ⎧ 1 for i = 1, ⎪ ⎪ ⎪ ⎪ 2 for i = 2, n − 1, ⎨⌈ ⌉ i for 3 ≤ i ≤ n − 2 when n = 6, 9, 12 ψ(u i vi ) = 2 ⎪ ⎪ ⎪ and for 3 ≤ i ≤ 2n + 2 when n ≥ 15, ⎪ 3 ⎩ k for 2n + 3 ≤ i ≤ n − 2 when n ≥ 15, 3 Please cite this article in press as: N. Hinding, et al., On irregularity strength of diamond network, AKCE International Journal of Graphs and Combinatorics (2017), https://doi.org/10.1016/j.akcej.2017.10.003.

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{

1 for i = 1, 2 for 2 ≤ i ≤ n − 2, ⎧ 1 for i = 1, ⎨ for i = n − 1, ψ(vi vi+1 ) = 2 ⎩n for 2 ≤ i ≤ n − 2. ⎧3 1 for i = n − 1, ⎪ ⎪ ⎪ i+1 ⎪ for i odd, 1 ≤ i < n − 1, when n = 6, 9 and for i odd, ⎪ ⎪ ⎨ 2 1 ≤ i ≤ 2n + 1, when n ≥ 12, 3 ψ(u i vi+1 ) = i + 1 for i even, 2 ≤ i < n − 1, when n = 6, 9 and for i ⎪ ⎪ 2 ⎪ ⎪ even, 2 ≤ i ≤ 2n , when n ≥ 12, ⎪ ⎪ 3 ⎩ k for 2n + 2 ≤ i ≤ n − 2 when n ≥ 12, 3

ψ(u i u i+1 ) =

Case 2. If n ≡ 1 (mod 3), n ≥ 7 the total labeling ψ : V (Brn ) ∪ E(Brn ) → {1, 2, . . . , k} is defined such that ψ(x) = 2, ⎧ ⎨1 ψ(vi ) = n+2 3 ⎩ n−1 3

for i = 1, n, for 2 ≤ i ≤ n+8 , 3 for n+11 ≤ i ≤ n − 1, 3

⎧ for i = 2, ⎪ ⎪k ⎨ 2 for i odd, 1 ≤ i < 2n−5 and for i = n, 3 ψ(xvi ) = 2n−2 3 for i even, 4 ≤ i ≤ , ⎪ ⎪ 3 ⎩ 2n+1 i − 2n−5 for ≤ i ≤ n − 1, 3 3 ⎧ , for 1 ≤ i ≤ 2n−2 ⎨1 3 2n−2 ψ(u i ) = i − 3 for 2n+1 ≤ i ≤ n − 2, 3 ⎩ 2 for i = n − 1, ⎧ 1 for i = 1, ⎪ ⎪ ⎨2 for i = n − 1, ψ(vi vi+1 ) = ⌈ n−1 ⌉ for 2 ≤ i ≤ n+5 , ⎪ 3 ⎪ ⎩ 3 k for n+8 ≤ i ≤ n − 2, 3 { 1 for i = 1, ψ(u i u i+1 ) = 2 for 2 ≤ i ≤ n − 2, ⎧ 1 for i = 1, ⎪ ⎪ ⎪ ⎪ for i = 2, n − 1, ⎨2⌈ ⌉ i for 3 ≤ i ≤ n − 2 when n = 7, 10 and for 3 ≤ i ≤ 2n+4 ψ(u i vi ) = 2 3 ⎪ ⎪ when n ≥ 13, ⎪ ⎪ ⎩ k for 2n+7 ≤ i ≤ n − 2 when n ≥ 13, 3 ⎧ 1 for i = n − 1, ⎪ ⎪ ⎨⌈ i+1 ⌉ for 1 ≤ i ≤ n − 2 when n = 7 and for 1 ≤ i ≤ 2n−2 2 3 ψ(u i vi+1 ) = when n ≥ 10, ⎪ ⎪ ⎩ k for 2n+1 ≤ i ≤ n − 2 when n ≥ 10. 3 Case 3. If n ≡ 2 (mod 3), n ≥ 8 we define the corresponding total labeling ψ : V (Brn ) ∪ E(Brn ) → {1, 2, . . . , k} as follows ψ(x) = 2, ⎧ ⎨1 n−2 ψ(vi ) = ⎩ 3 k

for i = 1, n, for 3 ≤ i ≤ n+4 , 3 for i = 2 and for

n+7 3

≤ i ≤ n − 1,

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ψ(xvi ) =

⎧ k ⎪ ⎪ ⎪ ⎨2

for i = 2, n − 1, for i odd, 1 ≤ i <

2n−7 and 3 2n−4 ≤ 3 ,

)

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for i = n,

3 for i even, 4 ≤ i ⎪ ⎪ ⎪ ⎩ 2n−1 i − 2n−7 for ≤ i ≤ n − 2, 3 3 { 1 for i = 1, ψ(u i u i+1 ) = 2 for 2 ≤ i ≤ n − 2, ⎧ for 1 ≤ i ≤ 2n−4 , ⎨1 3 2n−4 ψ(u i ) = i − 3 for 2n−1 ≤ i ≤ n − 2, 3 ⎩ 2 for i = n − 1, ⎧ 1 for i = 1, ⎪ ⎪ ⎨2 for i = 2, n − 1, ψ(u i vi ) = ⌈ i ⌉ for 3 ≤ i ≤ 2n−1 , ⎪ 3 ⎪ ⎩ 2 ≤ i ≤ n − 2, k for 2n+2 3 ⎧ for i = n − 1, ⎨1⌈ ⌉ i+1 for 1 ≤ i ≤ 2n−1 , ψ(u i vi+1 ) = 3 ⎩ 2 ≤ i ≤ n − 2, k for 2n+2 3 ⎧ for i = 1, ⎨1 for i = n − 1, ψ(vi vi+1 ) = 2 ⎩ k for 2 ≤ i ≤ n − 2.

For all three cases it is a matter of routine checking to verify that under the labeling ψ all vertex and edge labels are at most k and the weights of vertices of Brn are as follows: ⎧ for i = 1, ⎨4 for 2 ≤ i ≤ n − 2, wtψ (u i ) = 6 + i ⎩ 7 for i = n − 1, ⎧ 5 for i = 1, ⎪ ⎪ ⎪ ⎪ for 2 ≤ i ≤ k + 1, ⎨n + 3 + i for k + 2 ≤ i ≤ n − 2, wtψ (vi ) = n + 4 + i ⎪ ⎪ n + 5 + k for i = n − 1, ⎪ ⎪ ⎩ 6 for i = n. Clearly, the weights of vertices u i , 1 ≤ i ≤ n − 1, and vi , 1 ≤ i ≤ n, form a sequence of consecutive integers from 4 up to 2n + 2 and the weight of vertex x is greater than 2n + 4. It means that the weights of vertices of Brn are different for all pairs of distinct vertices. Thus the labeling ψ is the required vertex irregular total k-labeling. In fact, for every of previous three cases ⌈ ⌉ tvs(Brn ) ≤ n+1 . (5) 3 Combining (5) with the lower bound given in (4), we conclude that tvs(Brn ) = ⌈(n + 1)/3⌉ for n ̸= 5. □ 4. Conclusion In the paper we studied the total edge irregularity strength and the total vertex irregularity strength of diamond graphs Brn . We proved that tes(Brn ) = ⌈(5n − 3)/3⌉ for n ≥ 3, which is further support to the conjecture of Ivanˇco and Jendrol’. Moreover we proved that tvs(Brn ) = ⌈(n + 1)/3⌉ for n ≥ 3, n ̸= 5, and it is further support to the conjecture of Nurdin, Baskoro, Salman and Gaos. Acknowledgments The research for this article was supported by PPI-WCU Unhas 2017, APVV-15-0116, and by VEGA 1/0385/17. Please cite this article in press as: N. Hinding, et al., On irregularity strength of diamond network, AKCE International Journal of Graphs and Combinatorics (2017), https://doi.org/10.1016/j.akcej.2017.10.003.

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Please cite this article in press as: N. Hinding, et al., On irregularity strength of diamond network, AKCE International Journal of Graphs and Combinatorics (2017), https://doi.org/10.1016/j.akcej.2017.10.003.