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AKCE INTERNATIONAL JOURNAL OF GRAPHS AND COMBINATORICS

AKCE Int. J. Graphs Comb., 10, No. 2 (2013), pp. 147-155

On the total edge irregularity strength of generalized helm Diari Indriati Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Sebelas Maret, Surakarta, Indonesia e-mail: diari [email protected]

Widodo, Indah Emilia Wijayanti Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Gadjah Mada, Yogyakarta, Indonesia e-mail: [email protected], ind [email protected] and

Kiki Ariyanti Sugeng Department of Mathematics, Faculty of Mathematics and Science, University of Indonesia, Depok 16424, Indonesia e-mail: [email protected], [email protected]

AKCE Int. J. Graphs Comb., 10, No. 2 (2013), pp. 147-155

On the total edge irregularity strength of generalized helm Diari Indriati Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Sebelas Maret, Surakarta, Indonesia e-mail: diari [email protected]

Widodo, Indah Emilia Wijayanti Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Gadjah Mada, Yogyakarta, Indonesia e-mail: [email protected], ind [email protected] and

Kiki Ariyanti Sugeng Department of Mathematics, Faculty of Mathematics and Science, University of Indonesia, Depok 16424, Indonesia e-mail: [email protected], [email protected]

Abstract A total k-labeling is a map that carries vertices and edges of a graph G into a set of positive integer labels {1, 2, ..., k}. An edge irregular total k-labeling of a graph G is a total k-labeling such that the weights calculated for all edges are distinct. The weight of an edge uv in G is defined as the sum of the label of u, the label of v and the label of uv. The total edge irregularity strength of G, denoted by tes(G), is the minimum value of the largest label k over all such edge irregular total k-labelings. In this paper, we investigate the total edge irregularity strength of generalized helm, Hnm for n ≥ 3, m = 1, 2, and m ≡ 0 (mod 3).

Keywords: total k-labeling, edge irregular total k -labeling, total edge irregularity strength, generalized helm. 2010 Mathematics Subject Classification: 05C78.

1. Introduction Let G(V, E) be a connected, simple and undirected graph with vertex set V and edge set E. Wallis [10] defined a labeling (or valuation) of graph as follows: A labeling of a graph is a map that carries graph elements to the numbers (usually to the positive or non negative integers). The most common choices of domain are the set of all vertices and edges (such a labeling is called a total labeling), the vertex set alone (called a vertex labeling) or the edge set alone (called an edge labeling). In the recent development, the graph labeling is also defined as various functions (see Gallian [4]).

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On the total edge irregularity strength of generalized helm

For a graph G(V, E), Baˇca, et al. [3] defined a labeling f : V ∪ E → {1, 2, ..., k} to be a total k-labeling. An edge irregular total k-labeling of a graph G is a total k-labeling such that the weights calculated at all edges are distinct. The weight of an edge uv in G, denoted by wt(uv), is defined as the sum of the label of u, the label of v and the label of uv, that is wt(uv) = f (u) + f (uv) + f (v). They also defined the total edge irregularity strength of G, denoted by tes(G), as the minimum value of the largest label k over all such edge irregular total k-labelings. The total edge irregularity strength for various classes of graphs have been determined. For instances, Baˇca, et al. [3] gave a lower bound and an upper bound on total edge irregularity strength for any graph G with vertex set V and a non-empty edge set E, m l |E|+2 ≤ tes(G) ≤ |E|. They also gave a lower bound of total edge irregularity strength 3 m l ≤ tes(G). In the same paper, they proved for a graph G with maximum degree ∆, ∆+1 2

the total edge irregularity strength of paths, cycles, stars, wheels and friendship graphs, l m l m l m n+1 2n+2 , tes(K ) = , tes(W ) = for n ≥ 3, that are, tes(Pn ) = tes(Cn ) = n+2 1,n n 3 2 3 l m tes(Fn ) = 3n+2 respectively. 3

Furthermore, the total edge irregularity strength of tree G with edge set E and maximum degree ∆ had been found by Ivanˇco and Jendrol [7], that is tes(G) = m l mo nl |E|+2 , . According to this result, then the total edge irregularity strength max ∆+1 2 3 m l , as Siddiqui [9] had done for n ≥ 3 and of subdivision of stars, Snm , is (m+1)n+2 3

1 ≤ m ≤ 8. Nurdin et al. [8] proved the total edge irregularity strength of the corona product of paths with some graphs, namely paths, cycles, stars, gears, friendships and wheels. The total edge irregularity strength of the categorical product and strong product of two paths can be found in [1] and [2]. Haque [5] investigated the total edge irregularity strength of generalized Petersen graphs P (n, k) and proved that tes(P (n, k)) = n + 1 for l m k 6= n/2 and tes(P (n, k)) = 5n+4 for k = n/2. Indriati, et al. [6] determined the total 6

edge irregularity strength of helm, Hn , and disjoint union of t isomorphic helms, tHn , and found that tes(Hn ) = n + 1, while tes(tHn ) = tn + 1. In this paper, we investigate the total edge irregularity strength of generalized helm, Hnm for n ≥ 3, m = 1, 2 and m ≡ 0 (mod 3). 2. Main Results

A generalized helm, Hnm , is a graph obtained by inserting m vertices to every pendant edge of helm Hn . A generalized helm Hnm has (m+2)n+1 vertices and (m+3)n edges. Let the vertex set of Hnm be V (Hnm ) = {vi,j : 1 ≤ i ≤ n, 1 ≤ j ≤ m+1}∪{ui : 1 ≤ i ≤ n}∪{w} and the edge set of Hnm be E(Hnm ) = {(vi,j vi,j+1 ) : 1 ≤ i ≤ n, 1 ≤ j ≤ m} ∪ {(vi,m+1 ui ) :

Diari Indriati, Widodo, Indah Emilia Wijayanti and Kiki Ariyanti Sugeng

149

1 ≤ i ≤ n} ∪ {(ui ui+1(mod n) ) : 1 ≤ i ≤ n} ∪ {(wui ) : 1 ≤ i ≤ n}. Figure 1 illustrates the generalized helm Hnm .

v1,1 v1, 2 v1,m+1 u1

w

v 2 ,1

v 2, 2

v 2 ,m +1 u2

un

v n ,m +1 v n , 2

v n ,1

u3

v3 , m+1 v 3, 2 v 3,1 Figure 1: The Generalized Helm Hnm In the next theorem, we present the total edge irregularity strength of generalized helm Hn1 for n ≥ 3 as follows: m l . Theorem 2.1. For n ≥ 3, tes(Hn1 ) = 4n+2 3 Proof. From the lower bound of total edge irregularity strength we have that tes(Hn1 ) ≥ m l 4n+2 , n ≥ 3. To prove the equality, it is sufficient to show the existence of an edge 3 l m irregular total k1 -labeling with k1 = 4n+2 . 3

Let V (Hn1 ) = {ui , vi,1 , vi,2 : 1 ≤ i ≤ n} ∪ {w} be the vertex set and E(Hn1 ) = {wui , ui vi,2 , vi,2 vi,1 : 1 ≤ i ≤ n} ∪ {ui ui+1 : 1 ≤ i ≤ n − 1} ∪ {un u1 } be the edge set of Hn1 . Define a total labeling f1 in the following way: Case 1.

For n ≡ 0 (mod 3) and n ≡ 2 (mod 3); n ≥ 3.

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On the total edge irregularity strength of generalized helm

f1 (vi,1 ) = 1, i = 1, . . . , n,  l m i, for 1 ≤ i ≤ 2n+3 , 3 m l m f1 (vi,2 ) = l  2n+3 , for 2n+3 +1 ≤ i ≤ n, 3 3  m l n + 1, , for 1 ≤ i ≤ 2n+3 3 l m l m f1 (ui ) = i − 2n +n, for 2n+3 +1 ≤ i ≤ n, 3 3 l 4n + 2 m , f1 (w) = 3  l m 1, for 1 ≤ i ≤ 2n+3 , 3 l m l m f1 (vi,1 vi,2 ) = i − 2n , for 2n+3 +1 ≤ i ≤ n, 3 3

Case 2.

f1 (ui vi,2 ) = 1, i = 1, . . . , n,  l m 2n  i, for 1 ≤ i ≤   3 ,  l m l m 2n f1 (ui ui+1(mod n) ) = 2 2n 3m +1 − i, for 3 +1 ≤ i ≤ n − 1,  l   2n  for i = n, 3 +1, l m l m  2n−1 +i, for 1 ≤ i ≤ 2n+3 , 3 m 3 l m f1 (wui ) = l 2n+3 4n+2  , for +1 ≤ i ≤ n. 3 3

For n ≡ 1 (mod 3); n ≥ 3.

f1 (vi,1 ) = 1, i = 1, . . . , n,  l m i, for 1 ≤ i ≤ 2n 3 , l m f1 (vi,2 ) = l m 2n 2n  3 , for 3 +1 ≤ i ≤ n,  l m n + 1, for 1 ≤ i ≤ 2n 3 , l m l m f1 (ui ) = i − 2n +n + 1, for 2n +1 ≤ i ≤ n, 3 3 l 4n + 2 m , f1 (w) = 3  l m 1, for 1 ≤ i ≤ 2n 3 , l m l m f1 (vi,1 vi,2 ) = i − 2n +1, for 2n +1 ≤ i ≤ n, 3 3 f1 (ui vi,2 ) = 1, i = 1, . . . , n,

Diari Indriati, Widodo, Indah Emilia Wijayanti and Kiki Ariyanti Sugeng

  i,    l m f1 (ui ui+1(mod n) ) = 2 2n −1 − i,  l 3m    2n , 3 l m  2n +i, for 3 m f1 (wui ) = l 4n+2  , for 3

151

l m for 1 ≤ i ≤ 2n 3 −1, l m for 2n 3 ≤ i ≤ n − 1,

for i = n, l m 1 ≤ i ≤ 2n 3 −1, l m 2n 3 ≤ i ≤ n.

It can be seen that the function f1 is a map from V (Hn1 ) ∪ E(Hn1 ) into   4n+2 . Thus, f1 is a total k1 -labeling with k1 = 3

   1, 2, . . . , 4n+2 . 3

By observation, the weights of the edges are:

wt(vi,1 vi,2 ) = 2 + i, i = 1, . . . , n, wt(ui vi,2 ) = n + 2 + i, i = 1, . . . , n, wt(ui ui+1(mod n) ) = 2n + 2 + i, i = 1, . . . , n, wt(wui ) = 3n + 2 + i, i = 1, . . . , n. It can be seen that the weights of edges of Hn1 under the total k1 -labeling, f1 , form consecutive integers from 3 up to 4n + 2. It means that the weights of all  edges are 4n+2 distinct. So, the labeling is an edge irregular total k -labeling with k = . Therefore 1 1 3  4n+2  1 . tes(Hn ) = 3 Next, we continue to find the total edge irregularity strength of Hn2 . l m Theorem 2.2. For n ≥ 3, tes(Hn2 ) = 5n+2 . 3 Proof. From [3] we know that tes(Hn2 ) ≥

l

5n+2 3

m

, n ≥ 3. To prove the equality, next we m l . show the existence of an edge irregular total k2 -labeling with k2 = 5n+2 3

Let V (Hn2 ) = {vi,j : 1 ≤ i ≤ n, 1 ≤ j ≤ 3} ∪ {ui : 1 ≤ i ≤ n} ∪ {w} be the vertex set and E(Hn2 ) = {(vi,j vi,j+1 ) : 1 ≤ i ≤ n, 1 ≤ j ≤ 2} ∪ {(vi,3 ui ) : 1 ≤ i ≤ n} ∪ {(ui ui+1(mod n) ) : 1 ≤ i ≤ n} ∪ {(wui ) : 1 ≤ i ≤ n} be the edge set of Hn2 . Define a total labeling f2 in the following way: f2 (vi,1 ) = 1, i = 1, . . . , n, f2 (vi,2 ) = i, i = 1, . . . , n, f2 (vi,3 ) = n + 1, i = 1, . . . , n,

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On the total edge irregularity strength of generalized helm

 l m  n + i, for 1 ≤ i ≤ 2n+2 , 3 m l m l f2 (ui ) =  5n+2 , for 2n+2 +1 ≤ i ≤ n, 3 3 l 5n + 2 m , f2 (w) = 3 f2 (vi,1 vi,2 ) = f2 (vi,2 vi,3 ) = 1, i = 1, . . . , n,  m l  1, , for 1 ≤ i ≤ 2n+2 3 l m l m f2 (vi,3 ui ) =  i − 2n+2 +1, for 2n+2 +1 ≤ i ≤ n, 3 3  m l 2n−1  , n + 1 − i, for 1 ≤ i ≤  3   m l m l m l n−1 2n−1 2n−1 + 3 , for +1 ≤ i ≤ n − 1, i− f2 (ui ui+1(mod n) ) = 3  m3 l    4n−1 , for i = n, 3  l m l m 2n+2  4n+2 , for 1 ≤ i ≤ , 3 m l m l 3 m l f2 (wui ) =  4n+2 − 2n+2 +i, for 2n+2 +1 ≤ i ≤ n. 3 3 3

It can be seen that the function f2 is a map from V m l . Thus, f2 is a total k2 -labeling with k2 = 5n+2 3

(Hn2 ) ∪ E(Hn2 )

into



1, 2, . . . ,

l

5n+2 3

m .

Observe that

wt(vi,1 vi,2 ) = 2 + i, i = 1, . . . , n, wt(vi,2 vi,3 ) = n + 2 + i, i = 1, . . . , n, wt(vi,3 ui ) = 2n + 2 + i, i = 1, . . . , n, wt(ui ui+1(mod n) ) = 3n + 2 + i, i = 1, . . . , n, wt(wui ) = 4n + 2 + i, i = 1, . . . , n. So the weights of edges of Hn2 under the total k2 -labeling, f2 , form consecutive integers from 3 up to 5n + 2. It means that the weights of all edges are distinct. So, the labeling l m l m is an edge irregular total k2 -labeling with k2 = 5n+2 . Therefore tes(Hn2 ) = 5n+2 . 3 3 We give an example of an edge irregular total labeling of H42 in Figure 2. The following theorem shows the total edge irregularity strength of Hnm for m ≡ 0 (mod 3). Let m0 be the notation of m ≡ 0 (mod 3). l m Theorem 2.3. For n ≥ 3, tes(Hnm0 ) = (m0 +3)n+2 . 3

Diari Indriati, Widodo, Indah Emilia Wijayanti and Kiki Ariyanti Sugeng

153

1 1 1 1 5 1 5 4

5 6 6

1

2

5

1

1

6

8

8 6

1

5 1

4 1

1 1

6 3

2 7 1 5 1 3 1 1

Figure 2: An edge irregular total 8-labeling of H42 Proof. From the lower bound of total edge irregularity strength we have that tes(Hnm0 ) ≥ l m l m (m0 +3)n+2 m0 ) = (m0 +3)n+2 , next we show the existence , n ≥ 3. To prove that tes(H n 3 3 l m of an edge irregular total km0 -labeling with km0 = (m0 +3)n+2 . 3

Let V (Hnm0 ) = {vi,j : 1 ≤ i ≤ n, 1 ≤ j ≤ m0 + 1} ∪ {ui : 1 ≤ i ≤ n} ∪ {w} be the vertex set and E(Hnm0 ) = {(vi,j vi,j+1 ) : 1 ≤ i ≤ n, 1 ≤ j ≤ m0 } ∪ {(vi,m0 +1 ui ) : 1 ≤ i ≤ n} ∪ {(ui ui+1(mod n) ) : 1 ≤ i ≤ n} ∪ {(wui ) : 1 ≤ i ≤ n} be the edge set of Hnm0 . Define a total labeling fm0 as follows: For j odd ( j−1 0 for 1 ≤ i ≤ n; 1 ≤ j ≤ 2m 2 n + 1, m 3 + 3, l fm0 (vi,j ) = (m0 +3)n+2 0 , for 1 ≤ i ≤ n; 2m 3 3 + 3 < j ≤ m0 + 1.

For j even

fm0 (vi,j ) =

(

j−2 2m0 2 l 2 n + i, m for 1 ≤ i ≤ n; 2 ≤ j ≤ 3 + n + 2, (m0 +3)n+2 2 0 , for 1 ≤ i ≤ n; 2m 3 3 + n + 2 < j ≤ m0 + 1,

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On the total edge irregularity strength of generalized helm

fm0 (ui ) = fm0 (w) =

l

l

(m0 +3)n+2 3

(m0 +3)n+2 3

m

, i = 1, . . . , n,

m .

For m0 ≤ 6 and 1 ≤ i ≤ n, 1 ≤ j ≤ m0 fm0 (vi,j vi,j+1 ) = 1. For m0 > 6 fm0 (vi,j vi,j+1 ) =



1, (j −

2m0 +6 3

for 1 ≤ i ≤ n; 1 ≤ j ≤ 2m30 +6 , − 1)n + i, for 1 ≤ i ≤ n; 2m30 +6 < j ≤ m0 .

For m0 = 3 and 1 ≤ i ≤ n fm0 (vi,m0 +1 ui ) = 1. For m0 > 3 and 1 ≤ i ≤ n fm0 (vi,m0 +1 ui ) = fm0 (ui ui+1(mod n) ) = fm0 (wui ) =

m0 3 n

m0 −3 3

m0 − 6  n + i. 3

 n + i, for i = 1, . . . , n,

+ i, for i = 1, . . . , n.

It can be seen that the function fm0 is a map from V (Hnm0 ) ∪ E(Hnm0 ) into n l mo l m (m0 +3)n+2 (m0 +3)n+2 1, 2, . . . , . Thus, fm0 is a total km0 -labeling with km0 = . 3 3 Observe that

wt(vi,j vi,j+1 ) = (j − 1)n + 2 + i, i = 1, . . . , n; j = 1, . . . , m0 , wt(vi,m0 +1 ui ) = m0 n + 2 + i, i = 1, . . . , n, wt(ui ui+1(mod n) ) = (m0 + 1)n + 2 + i, i = 1, . . . , n, wt(wui ) = (m0 + 2)n + 2 + i, i = 1, . . . , n. So the weights of edges of Hnm0 under the total km0 -labeling, fm0 , form consecutive integers from 3 up to (m0 + 3)n + 2. It means that the weights of all edges are distinct. So, m l . Therefore the labeling is an edge irregular total km0 -labeling with km0 = (m0 +3)n+2 3 m l , for n ≥ 3. tes(Hnm0 ) = (m0 +3)n+2 3 3. Conclusion We have determined the total edge irregularity strength of generalized helm Hn1 , Hn2 , l l m m 5n+2 2) = and Hnm , and found that tes(Hn1 ) = 4n+2 , tes(H and tes(Hnm ) = n 3 3 l m (m+3)n+2 , for n ≥ 3 and m ≡ 0 (mod 3). 3

Diari Indriati, Widodo, Indah Emilia Wijayanti and Kiki Ariyanti Sugeng

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Furthermore, we conclude this paper with the following conjecture for the direction of further research which is still in progress. Conjecture 3.4. The total edge irregularity strength of generalized helm Hnm is tes(Hnm ) = l m (m+3)n+2 , for n ≥ 3 and m ≥ 1. 3 Acknowledgement The authors wish to thank the referees for their valuable suggestions and references, which helped to improve the paper.

References [1] A. Ahmad and M. Baˇ ca, Total edge irregularity strengths of the categorical product of two paths, Arc Combin., (To appear). [2] A. Ahmad, M. Baˇ ca, Y. Bashir and M. K. Siddiqui, Total edge irregularity strength of strong product of two paths, Arc Combin., 106 (2012), 449–459. [3] M. Baˇca, S. Jendrol’, M. Miller and J. Ryan, On irregular total labeling, Discrete Math., 307 (2007), 1378–1388. [4] J.A. Gallian, A Dynamic Survey of Graph Labeling, Electron. J. Combin., 19 (2012), # DS6. [5] K.M.M. Haque, Irregular total labelings of Generalized Petersen Graphs, Theory Comput Syst., 50 (2012), 537–544. [6] D. Indriati, Widodo, I.E. Wijayanti and K.A. Sugeng, Kekuatan Tak Regular Sisi Total pada Graf Helm dan Union Disjoinnya, Presented in Konferensi Nasional Matematika XVI, Unpad, Jatinangor, Jawa Barat, 3rd − 6th July, 2012. [7] J. Ivanˇco and S. Jendrol’, Total edge irregularity strength of trees, Discuss. Math. Graph Theory, 26 (2006), 449–456. [8] Nurdin, A.N.M. Salman and E.T. Baskoro, The total edge irregularity strengths of the corona product of paths with some graphs, J. Combin. Math. Combin. Comput., 65 (2008), 163–175. [9] M.K. Siddiqui, On edge irregularity strength of subdivision of star Sn , International Journal of Mathematics and Soft Computing, 2 (2012), 75–82. [10] W.D. Wallis, Magic Graphs, Birkhauser, Boston, 2001.