The Connected and Disjoint Union of Semi Jahangir Graphs Admit a Cycle-Super (a, d)- Atimagic Total Labeling

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS The Connected and Disjoint Union of Semi Jahangir Graphs Admit a Cycle-Super (a, d)- Atimagic Total Labeling Related content - On The Partition Dimension of a Lollipop Graph and a Generalized Jahangir Graph Maylinda Purna Kartika Dewi and Tri Atmojo Kusmayadi To cite this article: Dafik et al 016 J. Phys.: Conf. Ser View the article online for updates and enhancements. This content was downloaded from IP address on 1/01/018 at 06:9

2 ICMAME 015 Journal of Physics: Conference Series 693 (016) doi: / /693/1/01006 The Connected and Disjoint Union of Semi Jahangir Graphs Admit a Cycle-Super (a, d)-atimagic Total Labeling Dafik 1,,I.H.Agustin 1,3, D. Hardiyantik 3 1 CGANT - University of Jember Mathematics Education Department - University of Jember 3 Mathematics Department - University of Jember d.dafik@unej.ac.id; hestyarin@gmail.com Abstract. We assume that all graphs in this paper are finite, undirected and no loop and multiple edges. Given a graph G of order p and size q. Let H,H be subgraphs of G. By H -covering, we mean every edge in E(G) belongs to at least one subgraph of G isomorphic to a given graph H. A graph G is said to be an (a, d)-h-antimagic total labeling if there exist a bijective function f : V (G) E(G) P {1,,...,p+q} such that for all subgraphs H isomorphic to H, the total H-weights w(h) = f(v)+p v V (H ) e E(H ) f(e) form an arithmetic sequence {a, a+d, a+d,..., a+(s 1)d}, where a and d are positive integers and s is the number of all subgraphs H isomorphic to H. Such a labeling is called super if f : V (G) {1,,..., V (G) }. In this paper, we will discuss a cycle-super (a,d)-atimagicness of a connected and disjoint union of semi jahangir graphs. The results show that those graphs admit a cycle-super (a,d)-atimagic total labeling for some feasible d {0, 1,, 4, 6, 7, 10, 13, 14}. We use a handbook of graph theory written by Gross et. al [4] to define all basic definitions of graph in this paper. For p and q are respectively the order and size of graph, by a labeling of a graph, we mean any mapping that sends some set of graph elements to a set of positive integers. The labelings are called vertex labelings or edge labelings If the domain is respectively a vertex-set V (G) or a edge-set E(G). Moreover, the labelings are called total labelings if the domain is V (G) E(G). Simanjuntak et al. in [13] introduced an (a, d)-edge-antimagic total labeling of G of order p and size q. It is a oneto-one mapping f taking the vertices and edges of G onto {1,,...,p+ q} such that the edge-weights W f (uv) =f(u)+f(v)+f(uv),uv E(G) form an arithmetic sequence {a, a + d,...,a+(q 1)d}, where the first term a is a>0and the common difference d is d 0. Such a labeling is called super if the smallest possible labels appear on the vertices. Gutiérrez, and Lladó in [3, 8] expanded the edge-magic total labeling into a magic total covering. They defined that a graph G admits an H -magic covering, where H is subgraph of G isomorphic to a given graph H, if the total H-weights w(h) = v V (H ) f(v)+ e E(H ) f(e) =λ(h) is a constant magic sum and λ(h) is a constant supermagic sum of H if f : V (G) {1,,...,p}. Some relevant results can be found in [7, 9, 10, 1]. Recently Feňovčiková et. al [] proved that wheels are cycle antimagic. Motivated by these two previous labelings, Inayah et al. [5] introduced the (a, d) H- antimagic total labeling. A graph G is said to be an (a, d)-h-antimagic total labeling if there exist a bijective function f : V (G) E(G) {1,,...,p + q} such that for all subgraphs H isomorphic to H, the total H-weights w(h) = v V (H ) f(v) + e E(H ) f(e) form an arithmetic sequence Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 ICMAME 015 Journal of Physics: Conference Series 693 (016) doi: / /693/1/01006 {a, a + d, a +d,..., a +(s 1)d}, wherea and d are positive integers and s is the number of all subgraphs H isomorphic to H. Similarly, such a labeling is called super if f : V (G) {1,,...,p}. Inayah et. al [6] proved that, shack(h, k) which contains exactly k subgraphs isomorphic to H is H-super antimagic, for H is a non-trivial connected graph and k is an integer. We will discuss the existence of a cycle-super (a,d)-atimagicness of a connected and disjoint union of semi jahangir graphs. For H-supermagic graphs, we have found some results. For example Rizvi, et.al. [11] proved the disjoint union of isomorphic copies of fans, triangular ladders, ladders, wheels, and graphs obtained by joining a star K 1,n with K 1, and also disjoint union of non-isomorphic copies of ladders and fans are cycle-supermagic labelings, but for super antimagic labelings, it remains widely open to explore. The Results Prior to present the main results, we repropose a lemma proved by Dafik et.al in [1], it will be useful to find the existence of H-super antimagic graphs. This lemma showed a least upper bound for feasible value of d for a graph to be super (a, d)-h- antimagic total labeling. Lemma 1. [1] Let G be a simple graph of order p and size q. IfG is super (a, d)-h- antimagic total labeling then d (p G p H )p H +(q G q H )q H s 1,forH j are subgraphs isomorphic to H, p G = V (G), q G = E(G), p H = V (H ), q H = E(H ), and s = H j. Proof: Assume that a (p, q)-graph has a super (a, d)-h- antimagic total labeling f : V (G) E(G) {1,, 3,...,p G + q G } and the total H-weights w(h) = v V (H ) f(v)+ e E(H ) f(e) = {a, a + d, a +d,..., a +(s 1)d}. The minimum possible total H-weight in the labeling f is at least p H +(p G +1)+(p G +)+...+(p G + q H )= p H + p H + q H p G + q H + q H. Thus, a p H + p H + q H p G + q H + q H. On the other hand, the maximum possible total H-weight is at most p G + p G 1+p G (p G (p H 1)) + (p G + q G )+(p G + q G 1) + (p G + q G ) (p G + q G (q H 1)) = p H p G p H 1 (p H )+q H p G + q H q G q H 1 (q H ). So we obtain a +(s 1)d p H p G p H 1 (p H )+q H p G + q H q G q H 1 (q H ). Simplifying the inequality then we will have the desired upper bound of d. From now on we will introduce our terminology of connected semi Jahangir and disjoint union of semi Jahangir graphs. A semi Jahangir graph, denoted by SJ n, is a connected graph with vertex set V (SJ n ) = {p, x i,y k ;for 1 i n +1, 1 k n} and edge set E(SJ n ) = {px i ;1 i n +1} {x i y i ;1 i n} {y i x i+1 ;1 i n}. Since we study a super (a, d)-h- antimagic total labeling for H = C 4 isomorphic to H, thus p G = V (SJ n ) =n +, q G = E(SJ n ) =3n +1, p H = V (C 4 ) =4, q H = E(C 4 ) =4, s = H j = C 4 = n. If semi Jahangir graph SJ n has a super (a, d)-c 4 -antimagic total labeling then it follows from Lemma 1 the upper bound of d 0. A disjoint union of semi Jahangir graph, denoted by msj n, is a disconnected graph with vertex set V (msj n ) = {p j,x j i,yj k ;for 1 i n +1, 1 k n, 1 j m} and edge set E(mSJ n ) = {p j x j i ;1 i n +1, 1 j m} {xj i yj i ;1 i n, 1 j m} {y j i xj i+1 ;1 i n, 1 j m}. Since we study a super (a, d)-h- antimagic total labeling for H = C 4 isomorphic to H, thus p G = V (msj n ) =mn +m, q G = E(mSJ n ) =3mn + m, p H = V (C 4 ) =4, q H = E(C 4 ) =4, s = H j = C 4 = nm. If disjoint union of semi Jahangir graph msj n has a super (a, d)-f n -antimagic total labeling then it follows from Lemma 1 the upper bound of d 5. Now we start to describe the result of the super (a, d)-c 4 -antimagic total labeling of semi Jahangir graph, denoted by SJ n, in the following theorems. Theorem 1. For n, the graph SJ n admits a super (15n +1, 1) C 4 antimagic total labeling.

4 ICMAME 015 Journal of Physics: Conference Series 693 (016) doi: / /693/1/01006 Proof. Define the vertex and edge labeling f 1 as follows f 1 (p) = 1 f 1 (x i ) = i +1, for 1 i n +1 f 1 (y i ) = n + i +, for 1 i n f 1 (px i ) = n + i +, for 1 i n +1 f 1 (x i y i ) = 5n i +4, for 1 i n f 1 (y i x i+1 ) = 5n i +5, for 1 i n The vertex and edge labelings f 1 are a bijective function f 1 : V (SJ n ) E(SJ n ) {1,, 3,...,5n+3}. The H-weights of SJ n,for1 i n under the labeling f 1, constitute the following sets w f1 = f 1 (p)+f 1 (x i )+f 1 (x i+1 )+f 1 (y i )=(1)+(i+1)+(i+1+1)+(n+i+) = n+3i+6, and the total H- weights of SJ n constitute the following sets W f1 = w f1 +f 1 (px i )+f 1 (px i+1 )+f 1 (x i y i )+f 1 (y i x i+1 )= (n +3i +6)+(n + i +)+(n + i +1+)+(5n i +4)+(5n i +5)=15n + i +0.Itis easy to see that the set W f1 = {15n +1, 15n +,...,16n +0}. Therefore, the graph SJ n admits a super (15n +1, 1)-C 4 antimagic total labeling, for n. Theorem. For n, the graph SJ n admits a super (14n +, 7) C 4 antimagic total labeling. Proof. Define the vertex labeling f as f (p) =f 1 (p),f (x i )=f 1 (x i ),f (y i )=f 1 (y i ) and edge labeling f as follows f (px i ) = 4n + i +, for1 i n +1 f (x i y i ) = n + i +, for1 i n f (y i x i+1 ) = 3n + i +, for1 i n The vertex and edge labelings f are a bijective function f : V (SJ n ) E(SJ n ) {1,, 3,...,5n+3}. The H-weights of SJ n,for1 i n under the labeling f, constitute the following sets w f = w f1,and the total H-weights of SJ n ) constitute the following sets W f = w f +f (px i )+f (px i+1 )+f (x i y i )+ f (y i x i+1 )=(n+3i+6)+(4n+i+)+(4n+i+1+)+(n+i+)+(3n+i+) = 14n+7i+15. It is easy to see that the set W f = {14n +, 14n +9,...,1n +15}. Therefore, the graph SJ n admits a super (14n +, 7)-C 4 antimagic total labeling, for n. Theorem 3. For n, the graph SJ n admits a super (13n +3, 10) C 4 antimagic total labeling. Proof. Define the vertex and edge labeling f 3 as follows f 3 (p) = 1 f 3 (x i ) = i, for 1 i n +1 f 3 (y i ) = i +1, for 1 i n f 3 (px i ) = f (px i ) f 3 (x i y i ) = f (x i y i ) f 3 (y i x i+1 ) = f (y i x i+1 The vertex and edge labelings f 3 are a bijective function f 3 : V (SJ n ) E(SJ n ) {1,, 3,...,5n+3}. The H-weights of SJ n,for1 i n under the labeling f 3, constitute the following sets w f3 = f 3 (p)+f 3 (x i )+f 3 (x i+1 )+f 3 (y i ) = (1) + (i) +((i +1))+(i +1)=6i +4, and the total H- weights of SJ n constitute the following sets W f3 = w f3 +f 3 (px i )+f 3 (px i+1 )+f 3 (x i y i )+f 3 (y i x i+1 )= (6i +4)+(4n + i +)+(4n + i +1+)+(n + i +)+(3n + i +)=13n +10i +13.Itis easy to see that the set W f3 = {13n +3, 13n +33,...,3n +13}. Therefore, the graph SJ n admits a super (13n +3, 10)-C 4 antimagic total labeling, for n. 3

5 ICMAME 015 Journal of Physics: Conference Series 693 (016) doi: / /693/1/01006 Theorem 4. For n, the graph SJ n admits a super (11n +5, 13) C 4 antimagic total labeling. Proof. Define the vertex and edge labeling f 4 as follows f 4 (p) = 1 f 4 (x i ) = n + i +1, for 1 i n +1 f 4 (y i ) = n i +, for 1 i n f 4 (px i ) = n +3i, for 1 i n +1 f 4 (x i y i ) = n +3i +1, for 1 i n f 4 (y i x i+1 ) = n +3i +, for 1 i n The vertex and edge labelings f 4 are a bijective function f 4 : V (SJ n ) E(SJ n ) {1,, 3,...,5n+3}. The H-weights of SJ n,for1 i n under the labeling f 4, constitute the following sets w f4 = f 4 (p)+ f 4 (x i )+f 4 (x i+1 )+f 4 (y i )=(1)+(n+i+1)+(n+i+1+1)+(n i+) = 3n+i+6, and the total H- weights of SJ n constitute the following sets W f4 = w f4 +f 4 (px i )+f 4 (px i+1 )+f 4 (x i y i )+f 4 (y i x i+1 )= (3n + i +6)+(n +3i)+(n +3(i +1))+(n +3i +1)+(n +3i +)=11n +13i +1.Itis easy to see that the set W f4 = {11n +5, 11n +38,...,4n +1}. Therefore, the graph SJ n admits a super (11n +5, 13)-C 4 antimagic total labeling, for n. Theorem 5. For n, the graph SJ n admits a super ( 19n+54, 14) C 4 antimagic total labeling for n is even, and for n, the graph SJ n admits a super ( 19n+53, 14) C 4 antimagic total labeling for n is odd. Proof. Define the vertex and edge labeling f 5 as follows f 5 (p) = 1 f 5 (x i ) = i+3 n+i+4, for 1 i n +1;iis odd, for 1 <i<n+1;iis even,nis even n+i+3, for 1 <i n +1;iis even,n is odd f 5 (y i ) = n + i +, for 1 i n f 5 (px i ) = f 4 (px i ) f 5 (x i y i ) = f 4 (x i y i ) f 5 (y i x i+1 ) = f 4 (y i x i+1 ) The vertex and edge labelings f 4 are a bijective function f 5 : V (SJ n ) E(SJ n ) {1,, 3,...,5n+3}. The H-weights of SJ n,for1 i n under the labeling f 5, constitute the following sets w f5 = f 5 (p)+f 5 (x i )+f 5 (x i+1 )+f 5 (y i )=1+( i+3 )+(n+i+1+4 )+(n +i +4)= 3n+4i+14 for even n, w f5 = f 5 (p)+f 5 (x i )+f 5 (x i+1 )+f 5 (y i )=1+( i+3 )+(n+i+1+3 )+(n +i +4)= 3n+4i+14 for odd n and the total H-weights of SJ n constitute the following sets W f5 = w f5 + f 5 (px i )+f 5 (px i+1 )+ f 5 (x i y i )+f 5 (y i x i+1 )=( 3n+4i+14 )+(n +3i)+(n +3(i +1))+(n +3i +1)+(n +3i +)= 19n+8i+6 for even n and W f5 = w f5 + f 5 (px i )+f 5 (px i+1 )+f 5 (x i y i )+f 5 (y i x i+1 )=( 3n+4i+13 )+ (n +3i)+(n +3(i +1))+(n +3i +1)+(n +3i +)= 19n+8i+5 for odd n. It is easy to see that the set W f5 = { 19n+54, 19n+8,..., 47n+6 } for even n and W f5 = { 19n+53, 19n+81,..., 47n+5 } for odd n. Therefore, the graph SJ n admits a super ( 19n+54, 14) C 4 antimagic total labeling for n with even n. And the graph SJ n admits a super ( 19n+53, 14) C 4 antimagic total labeling for n with odd n. We continue to show the result of the super (a, d)-c 4 -antimagic total labeling of disjoint union of semi Jahangir graph, SJ n, in the following theorems. 4

6 ICMAME 015 Journal of Physics: Conference Series 693 (016) doi: / /693/1/01006 Theorem 6. For m, n, the graph msj n admits a super (18mn +14m +4, 0)-C 4 antimagic total labeling. Proof. For 1 j m,define the vertex and edge labeling g 1 as follows g 1 (p j ) = j, 1 j m g 1 (x j i ) = mi + j m, for 1 i n +1, 1 j m g 1 (y j i ) = mn mi +3m j +1, for 1 i n, 1 j m g 1 (p j x j i ) = 4mn + mi + m + j, for 1 i n +1, 1 j m g 1 (x j i yj i ) = 4mn mi +4m j +, for 1 i n, 1 j m g 1 (y j i xj i+1 ) = 4mn mi +4m j +1, for 1 i n, 1 j m The vertex and edge labelings g 1 are a bijective function g 1 : V (msj n ) E(mSJ n ) {1,, 3,...,5mn +3m}. TheH-weights of msj n,for1 i n and 1 j m under the labeling g 1, constitute the following sets w g1 = g 1 (p j )+g 1 (x j i )+g 1(x j i+1 )+g 1(y j i )=(j)+(mi + j m)+ (m(i+1)+j m)+(mn mi+3m j +1) = mn+mi+3m+j +1, and the total H-weights of msj n constitute the following sets W g1 = w g1 + g 1 (p j x j i )+g 1(p j x j i+1 )+g 1(x j i yj i )+g 1(y j i xj i+1 )= (mn +mi +3m +j +1)+(4mn + mi + m + j) +(4mn + m(i +1)+m + j) +(4mn mi +4m j +)+(4mn mi +4m j +1)=18mn +14m +4. It is easy to see that the set W g1 = {18mn +14m +4, 18mn +14m +4,...,18mn +14m +4}. Therefore, the graph msj n admits a super (18mn +14m +4, 0)-C 4 antimagic total labeling, for m, n. Theorem 7. For m, n, the graph msj n admits a super (17mn +14m +5, )-C 4 antimagic total labeling. Proof. For 1 j m, define the vertex labeling g as g (p j )=g 1 (p j ),g (x j i )=g 1(x j i ),g (y j i )= g 1 (y j i ) and edge labeling g as follows g (p j x j i ) = 4mn + mi +m j +1, for 1 i n +1, 1 j m g (x j i yj i ) = 3mn mi +m + j, for 1 i n, 1 j m g (y j i xj i+1 ) = 4mn mi +m + j, for 1 i n, 1 j m The vertex and edge labelings g 1 are a bijective function g : V (msj n ) E(mSJ n ) {1,, 3,...,5mn +3m}. The H-weights of msj n,for1 i n and 1 j m under the labeling g, constitute the following sets w g = w g1, and the total H-weights of msj n constitute the following sets W g = w g + g (p j x j i )+g (p j x j i+1 )+g (x j i yj i )+g (y j i xj i+1 ) = (mn +mi + 3m +j +1)+(4mn + mi +m j +1)+(4mn + m(i +1)+m j +1)+(3mn mi + m + j) +(4mn mi +m + j) =17mn +mi +1m +j +3. It is easy to see that the set W g = {17mn +14m +5, 17mn +14m +7,...,19mn +14m +3}. Therefore, the graph msj n admits a super (17mn +14m +5, )-C 4 antimagic total labeling, for m, n. Theorem 8. For m, n, the graph msj n admits a super (16mn +14m +6, 4)-C 4 antimagic total labeling. Proof. For 1 j m, define the vertex labeling g 3 as g 3 (p j )=g 1 (p j ),g 3 (x j i )=g 1(x j i ),g 3(y j i )= 5

7 ICMAME 015 Journal of Physics: Conference Series 693 (016) doi: / /693/1/01006 g 1 (y j i ) and edge labeling g 3 as follows g 3 (p j x j i ) = 4mn + mi + m + j; 1 i n +1, 1 j m, dan i ganjil g 3 (p j x j i ) = 4mn + mi +m j +1; 1 i n +1, 1 j m, dan i genap g 3 (x j i yj i ) = mn + mi + m + j, for 1 i n, 1 j m g 3 (y j i xj i+1 ) = 4mn mi +m + j, for 1 i n, 1 j m The vertex and edge labelings g 1 are a bijective function g 3 : V (msj n ) E(mSJ n ) {1,, 3,...,5mn +3m}. The H-weights of msj n,for1 i n and 1 j m under the labeling g 3, constitute the following sets w g3 = w g1, and the total H-weights of msj n constitute the following sets W g3 = w g3 + g 3 (p j x j i )+g 3(p j x j i+1 )+g 3(x j i yj i )+g 3(y j i xj i+1 ) = (mn +mi + 3m +j +1)+(4mn + mi + m + j) +(4mn + m(i +1)+m j +1)+(mn + mi + m + j) +(4mn mi +m + j) =16mn +4mi +10m +4j +. It is easy to see that the set W g3 = {16mn +14m +6, 16mn +14m +10,...,0mn +14m +}. Therefore, the graph msj n admits a super (16mn +14m +6, 4)-C 4 antimagic total labeling, for m, n. Theorem 9. For m, n, the graph msj n admits a super (15mn +14m +7, 6)-C 4 antimagic total labeling. Proof. For 1 j m, define the vertex labeling g 4 as g 4 (p j )=g 1 (p j ),g 4 (x j i )=g 1(x j i ),g 4(y j i )= g 1 (y j i ) and edge labeling g 4 as follows g 4 (p j x j i ) = 4mn + mi + m + j, for 1 i n +1, 1 j m g 4 (x j i yj i ) = mn + mi + m + j, for 1 i n, 1 j m g 4 (y j i xj i+1 ) = 3mn + mi + m + j, for 1 i n, 1 j m The vertex and edge labelings g 4 are a bijective function g 4 : V (msj n ) E(mSJ n ) {1,, 3,...,5mn+3m}. TheH-weights of msj n,for1 i n and 1 j m under the labeling g 4, constitute the following sets w g4 = w g1, and the total H-weights of msj n constitute the following sets W g4 = w g4 +g 4 (p j x j i )+g 4(p j x j i+1 )+g 4(x j i yj i )+g 4(y j i xj i+1 )=(mn+mi+3m+j +1)+(4mn+ mi+m+j)+(4mn+m(i+1)+m+j)+(mn+mi+m+j)+(3mn+mi+m+j) =15mn+6mi+8m+ 6j +1. It is easy to see that the set W g4 = {15mn+14m+7, 15mn+14m+13,...,1mn+14m+1}. Therefore, the graph msj n admits a super (15mn +14m +7, 6)-C 4 antimagic total labeling, for m, n. Concluding Remarks A least upper bound of difference d for connected and disjoint union of graphs are respectively d 0 and d 5. Apart from obtained d above, we haven t found any result yet, so we propose the following open problem: Open Problem 1. Apart from d {1, 7, 10, 13, 14}, determine a super (a, d) C 4 -antimagic total labeling of connected SJ n,ford 0 and n. Open Problem. Apart from d {0,, 4, 6}, determine a super (a, d) C 4 -antimagic total labeling of disjoint union of m copies of SJ n,ford 5 and m, n. Acknowledgement. We gratefully acknowledge the support from DPM Fundamental Research Grant DIKTI 015 and CGANT - University of Jember. 6

8 ICMAME 015 Journal of Physics: Conference Series 693 (016) doi: / /693/1/01006 References [1] Dafik, Slamin, Wuria Novitasari, Super (a,d)-h- antimagic total covering of shackle graph, Indonesian Journal of Combinatorics (015), submitted [] A. S. Feňovčiková, M. Baca, M. Lascsáková, M. Miller, J. Ryan, Wheels are Cycle-Antimagic, Electronic Notes in Discrete Mathematics 48 (015), 1118 [3] A. Gutiérrez, and A. Lladó, Magic Coverings, J. Combin. Math. Combin. Comput 55 (005), [4] J.L. Gross, J. Yellen and P. Zhang, Handbook of Graph Theory, Second Edition, CRC Press, Taylor and Francis Group, 014 [5] N. Inayah, A.N.M. Salman and R. Simanjuntak, On (a, d) H-antimagic coverings of graphs, J. Combin. Math. Combin. Comput. 71 (009), [6] N. Inayah, R. Simanjuntak, A. N. M. Salman, Super (a, d) H-antimagic total labelings for shackles of a connected graph H, The Australasian Journal of Combinatorics, 57 (013), [7] P. Jeyanthi, P. Selvagopal, More classes of H-supermagic Graphs, Intern. J. of Algorithms, Computing and Mathematics 3(1) (010), [8] A. Lladó and J. Moragas, Cycle-magic graphs, Discrete Math. 307 (007), [9] T.K. Maryati, A. N. M. Salman, E.T. Baskoro, J. Ryan, M. Miller, On H- supermagic labelings for certain shackles and amalgamations of a connected graph, Utilitas Mathematica, 83 (010), [10] A. A. G. Ngurah, A. N. M. Salman, L. Susilowati, H-supermagic labeling of graphs, Discrete Math., 310 (010), [11] S.T.R. Rizvi, K. Ali, M. Hussain, Cycle-supermagic labelings of the disjoint union of graphs, Utilitas Mathematica, (014), in press. [1] M. Roswitha, E.T. Baskoro, H-magic covering on some classes of graphs, American Institute of Physics Conference Proceedings 1450 (01), [13] R.Simanjuntak, M.Miller and F.Bertault, Twonew (a, d)-antimagic graph labelings, Proc. Eleventh Australas. Workshop Combin. Alg. (AWOCA) (000),