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AKCE International Journal of Graphs and Combinatorics I (IUI) III—I

AKCE International Journal of Graphs and Combinatorics

www.elsevier.com/locate/akcej

On the edge irregularity strength of corona product of cycle with

isolated vertices

I. Tarawneha, R. Hasnia'*, A. Ahmadb

a School of Informatics and Applied Mathematics, University Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia b College of Computer Science & Information Systems, Jazan University, Jazan, Saudi Arabia

Received 7 February 2016; received in revised form 8 June 2016; accepted 16 June 2016

Abstract

In this paper, we investigate the new graph characteristic, the edge irregularity strength, denoted as es, as a modification of the well known irregularity strength, total edge irregularity strength and total vertex irregularity strength. As a result, we obtain the exact value of an edge irregularity strength of corona product of cycle with isolated vertices.

© 2016 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: Irregularity assignment; Irregularity strength; Edge irregularity strength; Unicyclic graphs

1. Introduction

Let G be a connected, simple and undirected graph with vertex set V (G) and edge set E (G). By a labeling we mean any mapping that maps a set of graph elements to a set of numbers (usually positive integers), called labels. 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) U E(G), then we call the labeling total labeling. Thus, for an edge k-labeling 8 : E(G) ^ {1, 2,..., k} the associated weight of a vertex x € V(G) is

ws(x) = 8(xy),

where the sum is over all vertices y adjacent to x.

Chartrand et al. [1] introduced edge k-labeling 8 of a graph G such that w8(x) = J2 8(xy) for all vertices x, y € V (G) with x = y. Such labelings were called irregular assignments and the irregularity strength s (G) of a graph G is known as the minimum k for which G has an irregular assignment using labels at most k. This parameter has attracted much attention [2-8].

Peer review under responsibility of Kalasalingam University.

* Corresponding author. Fax: +60 96694660. E-mail addresses: ibrahimradi50@yahoo.com (I. Tarawneh), hroslan@umt.edu.my (R. Hasni), ahmadsms@gmail.com (A. Ahmad).

http://dx.doi.org/10.1016Zj.akcej.2016.06.010

0972-8600/©© 2016 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/).

2 I. Tarawneh et al. / AKCE International Journal of Graphs and Combinatorics I ^111) Ill-Ill

Motivated by these papers, Baca et al. [9] defined a vertex irregular total k-labeling of a graph G to be a total labeling of G, ^ : V (G) U E(G) ^ {1, 2, ..., k}, such that the total vertex-weights

wt (x) = ^(x) + ^^ xy)

xye E (G)

are different for all vertices, that is, wt (x) = wt (y) for all different vertices x, y € V (G). The total vertex irregularity strength of G, tvs(G), is the minimumk for which G has a vertex irregular total k-labeling. They also defined the total labeling ^ : V(G) U E(G) ^ {1, 2, ..., k} to be an edge irregular total k-labeling of the graph G if for every two different edges xy and x'y' of G one has wt (xy) = ) + ty(xy) + ^ (y) = wt (x'y') = ') + 'y') + ^ (y'). The total edge irregularity strength, tes (G), is defined as the minimum k for which G has an edge irregular total k-labeling. Some results on the total vertex irregularity strength and the total edge irregularity strength can be found in [10-16,8,17-19].

The most complete recent survey of graph labelings is [20].

A vertex k-labeling 0 : V (G) ^ {1, 2, ..., k} is called an edge irregular k-labeling of the graph G if for every two different edges e and f, there is w^(e) = w^(f), where the weight of an edge e = xy € E(G) is w<p (xy) = <p(x) + 0(y). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es (G) (see [21]).

In [21], the authors estimated the bounds of the edge irregularity strength es and then determined its exact values for several families of graphs namely, paths, stars, double stars and Cartesian product of two paths. Mushayt [22] determined the edge irregularity strength of cartesian product of star, cycle with path P2 and strong product of path Pn with P2. Tarawneh et al. [23] investigated the edge irregularity strength of corona product of graph with paths. Recently, Ahmad [24] determined the exact value of the edge irregularity strength of corona graph Cn O K1 (or sun graph Sn).

The following theorem established lower bound for the edge irregularity strength of a graph G.

Theorem 1 ([21]). Let G = (V, E) be a simple graph with maximum degree A = A(G). Then IE(G)| + 1 ~

es (G ) > max

A(G ) .

In this paper, we determine the exact value of edge irregularity strength of corona graphs Cn O mK 1, m > 2. 2. Two lemmas

The corona product of two graphs G and H, denoted by, G O H, is a graph obtained by taking one copy of G (which has n vertices) and n copies H1, H2,..., Hn of H, and then joining the i th vertex of G to every vertex in Hi. The corona product Cn O mK 1 is a graph with the vertex set V(Cn O mK 1) = {x,-, yj : 1 < i < n, 1 < j < m}

and edge set E(Cn O mK 1) = {x^-+1 : 1 < i < n — 1} U {xiyJi : 1 < i < n, 1 < j < m} U {xnx1} is the edge set of Cn O mK 1. The corona product Cn O K1 is also know as sun graph Sn. Ahmad [24] determined the exact value of the edge irregularity strength of Cn O K1 = Sn.

The following two lemmas determine the exact value of the edge irregularity strength for two particular cases.

Lemma 1. Let Cn O 2K1, n > 3, be a corona graph. Then, es (Cn O 2K1) = [3n+11.

Proof. The graph Cn O 2K1 has 3n vertices and 3n edges. The maximum degree of A(Cn O 2K1) is 4. Therefore, by Theorem 1, we have that es(Cn O 2k 1) > max

3m+1 2

= \ 3n+1 ]. To prove the equality, it suffices to

prove the existence of an optimal edge irregular [1—labeling. Assume k = {1, 2, ..., T 1} be the vertex labeling such that:

.Let 0i : V (Cn O 2 K i) ^

For l < i < L § J + l, <Pi(xt ) = 2f1 + T 21 and for L f J + 2 < i < n, 0i (xt ) = k - 2fn— 1 - L ^ & )

2L—J + L 2 J + j fori < i < L 2 J, 1 < j < 2. For i = L | J + 1, n = 1 (mod4), 1 < j < 2, 0i (yj ) = ^ + 3 j.

I. Tarawneh et al. / AKCE International Journal of Graphs and Combinatorics I illll) III—I

For i = L2J + 1, n = 3 (mod4), 1 < j < 2, 01 (yj) = 31-9 + 2j. For i = 2 + 1, n = 0 (mod2), 1 < j < 2, 01 (yj) = 2L^J + L2J + j + 1 and L|J + 1 < i < n, 1 < j < 2, 01 (yj) = k - 2rn-2±i 1 - f^1 + j.

,j\ — I i-1

The weights of the edges are as follows:

W01 (XiXi + 1 )

3i + 1, k+2 2k - 3(n

~i - 1" n-i-1 + i n - i - 1 , if i = n

2 2 2 _ 2 _ L 2 J

i ) + 1,

n -2 +1

+ 1 < i < n - 1.

w01 (xnx1 ) = k + 1. For 1 < i < L2J, 1 < j < 2, w01 (x^j) = 3i + j - 2. For n = 1 (mod4), 1 < j <

2, W01 (xl»J+1 y^iJ+1 ) = ^IT5 + 3j. If n = 3 (mod4) and 1 < j < 2, then w01 (xl»j+1 y[nJ+1) = i + 2j.

3(n-1)

If n = 0 (mod2) and 1 < j < 2, then w01 (xLi J+1 y^J+1) = ^^ + j. For L2J + 1 < i < n, 1 < j < 2

W01 (xiyi) = 2k - 3(n - i) + j - 2.

We can see that all vertex labels are at most

. The edge weights under the labeling 01 successively attain values {2, 3, ..., 2k}. Thus the edge weights are distinct for all pairs of distinct edges and the labeling 01 provides the

upper bound i.e. es(Cn O 2Ki) < [1. Combining with lower bound, we get that es (Cn O 2Ki) = [].

This completes the proof. □

Lemma 2. Let Cn O 3K1, n > 3, be a corona graph. Then, es (Cn O 3K1 ) = f

4n+1-\

Proof. The graph Cn O 3Ki has 4n vertices and 4n edges. The maximum degree of A(Cn O 3Ki) is 5. By Theorem i,

4w+1 2

we have that es (Cn O 3K1) > max

an optimal edge irregular [4n++1 ]-labeling. Assume k = the vertex labeling such that:

, 5 [ = [ 4n±i ]. To prove the equality, it suffices to prove the existence of

.Let 02 : V (C„ O 3 K1) ^ {1, 2,..., f ^ 1} be

For 1 < i < L2J + 1, 02(xi) = 3f^ 1 + f21 and for L2J+ 2 < i < n, 02(x) = k - 3f^ 1 -L1-1 J. 02(yj) =

3L^J + L2J + j for 1 < i < L2J, 1 < j < 3. For i = L2J + 1, n = 1 (mod4), 1 < j < 2, 02(y/) = n + j - 1. 02(y3) = n + 4, for i = L2J + 1, n = 1 (mod4). For i = L2J + 1, n = 3 (mod4), 1 < j < 2, 02(yj) = n + j - 2 and 02(yf) = n + 2, for i = L2J + 1, n = 3(mod4). For i = L2J + 1, n = 0 (mod2), 1 < j < 3, 01(yj) =

3L^J + L2J + j + i and L2J + i < i < n, i < j < 3, 0i(yj) = k - 3T The weights of the edges are as follows:

n-i+1 -1 2 1

f n-1+ j.

W02 (xi xi + 1 )

4i + 1, k+3

~i - 1" n-i-1 + i n - i - 1 , if i = n

2 2 2 _ 2 _ L 2 J

2k - 4(n - i) + 1,

n -2 +1

+ 1 < i < n

W02(xnxi) = k + i. For i < i < L2J, i < j < 3, W02(xiyi) = 4i + j — 3. For n odd, i < j < 2, W02(xl2j+iyjnj+i) = 2n + j — i. If n = i (mod4), then W02(xl»j+iy3«J+i) = 2n + 4. If n = 0 (mod2) and i < j < 3, then W02(xl«j+iyjnj+i) = 2n + j + 2. If n = 3 (mod4), then W02(xl«j+iy3^j+i) = 2n + 3. For

L|J + 1 < i < n - 1, 1 < j < 3, W02(xiyj) = 2k - 4(n - i) + j - 3.

We can see that all vertex labels are at most

. The edge weights under the labeling 02 attain distinct values

from 2 up to 2k. Thus the edge weights are distinct for all pairs of distinct edges and the labeling 02 provides the

This completes the proof. □

upper bound i.e. es (Cn O 3Ki) < [4n+i 1. Combining with lower bound, we get that es (Cn O 3Ki) = [1.

I. Tarawneh et al. / AKCE International Journal of Graphs and Combinatorics I illll) III—I

3. Main result

In this section, we determine the exact value of the edge irregularity strength of corona product Cn O mKi, m ^ 2.

Theorem 2. Let Cn O mKi be a corona graph with n > 3 and m > 2. Then es(Cn O mKi) =

'n (m + 1) + 1

Proof. The graph Cn O mKi has (m + i)n vertices and (m + i)n edges. The maximum degree of Cn O mKi, A(Cn O mK i) is m + 2. Therefore, by Theorem i, we have that es (Cn O mK i) > max { ^mn+n+i , m + 2j =

mn+n+1

1. For m = 2 and m = 3, see Lemmas i and 2, respectively. To prove the equality for m > 4, it

. Let 03 :

suffices to prove the existence of an optimal edge irregular Tmn+n+i]—labeling. Assume k =

V(Cn O mK 1 )

{1, 2, ..., r mn+n+1 ]} be the vertex labeling such that:

mn+n+1 2

For 1 < i < L2J + 1, 03(xi) = m f1 + T21 and for L2J + 2 < i < n, 03(xt) — k - mfV1 - LVJ.

2j + i, ^ — I 2 I + I 2 I L2J + ^ < 1 < vov*u = "H 2 1 L 2

03(yj) — mLJ + L2J + j for 1 < i < L2J, 1 < j < m and i — L2J + 1, n = 1 (mod2), 1 < j < ff 1. For

i — L2J + 1, « = 0 (mod2), 1 < j < m, (03(y}) — m L^J + L2J + j + 1 and 03(yj) — m LJ + L2 J ^ j + 2 for i — L 2 J + 1, n = 1 (mod 2), f f 1 + 1 < j < m.

03 (yj) —

n-i+1 n - i

n-i+1 n - i

+ j - 1, if i —

and 1 < j < + j, otherwise.

+ 2 for n = 3 (mod 4) m — 3

The weights of the edges are as follows: (m + 1)i + 1,

(XiXi + 1)

if 1 i

~i - 1" n - i - 1 + i n-i-1

2 2 2 _ 2 _

2k - (m + 1)(n - i) + 1,

if i — n

n .2 +1

+ 1 < i < n - 1

W03(xnxi) = k + i. For i < i < L2J, i - j - m, (xi_yiJ) = (m + i)i + j — m. For n odd, i < j < T f 1, W03 (xl 2 j+i yL» J+i) = (m + i)(L 2 J + i) + j — m .If n = i (mod4) and i < j < T f 1, then W03(xLnJ+iyjnJ+i) = (m + i)(L2J + i) + j — m + 2. For n even, i < j < m and n = 3 (mod4), Tmr 1 + i < j <

m, W03(xl2J+1 y[nJ+1 ) — (m + 1)(L2J + 1) + j - m + 1.

W03 (Xiyi ) —

2k - (m + 1)(n - i) + j

for i — 1 < j <

2k - (m + 1)(n - i ) + j - m,

+ 2, n = 3 (mod 4), m3

otherwise.

We can see that all vertex labels are at most

mra+ra+1

. The edge weights under the labeling 03 successively attain values {2, 3, ..., 2k}. Thus the edge weights are distinct for all pairs of distinct edges and the labeling 03 provides the upper bound on es(Cn O mKi) < T mnH2n+i 1. Combining with lower bound, we get that es (Cn O mKi) — r mn+n+i This completes the proof. □

fmn+n+11

I. Tarawneh et al. / AKCE International Journal of Graphs and Combinatorics I ^111) Ill-Ill 5

4. Conclusion

In this paper, we discussed the new graph characteristic, the edge irregularity strength es, as a modification of the well-known irregularity strength, total edge irregularity strength and total vertex irregularity strength (see [2i,24,22, 23]). We obtained the exact values for edge irregularity strength of corona graphs Cn O mKi, m > 2 and n > 3.

Acknowledgment

The authors would like to thank the referee for his/her valuable comments. References

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