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  n On P ⋄ P -Supermagic Labeling of Edge Corona

  2 Product of Cycle and Path Graph

  Riza Yulianto and Titin Sri Martini

  Mathematics Department of Mathematics and Natural Sciences Faculty, Universitas Sebelas Maret, Surakarta, Indonesia E-mail: yuliantoriza48@gmail.com, titinsmartini@gmail.com Abstract.

  A simple graph G = (V, E) admits a H-covering, where H is subgraph of G, if every edge in E belongs to a subgraph of G isomorphic to H. Graph G is H-magic if there is a total ∪

  ′ ′ ′ E

  , E labeling f : V (G) (G) → 1, 2, ..., |V (G)| + |E(G)|, such that each subgraph H = (V ) of def

  G = f f isomorphic to H and satisfying f (H ) Σ vϵV ′ (v) + Σ eϵE ′ (e) = m(f ) where m(f ) is a con- stant magic sum. Additionaly, G admits H-supermagic if f (V ) = 1, 2, ..., |V |. The edge corona C n n n n n

  ⋄ P of C and P is defined as the graph obtained by taking one copy of C and n copies of P n n n , and then joining two end-vertices of the i-th edge of C to every vertex in the i-th copy of P .

  This research aim is to find H-supermagic covering on an edge corona product of cycle and path n n 2 n graph C ⋄ P where H is P ⋄ P . We use k-balanced multiset to solve our reserarch. Here, we 2 n n n find that an edge corona product of cycle and path graph C ⋄P is P ⋄P supermagic for n ≥ 3.

  1. Introduction Let G be a simple graph G = (V, E), where V is a set of vertices, and E is a set of edges. Chartrand and Lesniak [1] defined that cycle graph is a circuit with no repeated vertices, except the first and last vertices. The cycle graph with n vertices is denoted by C n . They also defined path graph is a walk with no repeated vertices, path graph with n vertices is denoted by P n .

  Let G 1 and G 2 are two graphs on disjoint sets of n 1 and n 2 vertices, m 1 and m 2 edges, respectively. The edge corona G 1 ⋄ G 2 is defined as the graph obtained by taking one copy of

  G and m copies of G , and then joining two end-vertices of the i-th edge of G to every vertex

  1

  1

  2

  1 in the i-th copy of G . Note that the edge corona G ⋄ G of G and G has n + m n vertices

  2

  1

  2

  1

  2

  1

  1

  2 and m + 2m n + m m edges, for detail defnition of graph see [4].

  1

  1

  2

  1

  2 Gallian [2] defined a graph labeling as an assignment of integers to the vertices or edges, or both, subject to certain condition. Magic labelings was first introduced in 1963 by Sedl´aˇck [9]. The concept of H-magic graphs was introduced in [3]. An edge-covering of a graph G is a family of different subgraphs H , H , ..., H such that each edge of E belongs to at least

  1 2 k one of the subgraphs H i , 1 ≤ i ≤ k. Then, it is said that G admits an (H , H , ..., H k )-

  1

  2 edge covering. If every H i is isomorphic to a given graph H, then we say that G admits an H-covering. Suppose that G = (V (G), E(G)) admits an H-covering. A bijective function f

  : V (G)∪E(G) → {1, 2, ..., |V (G)|+|E(G)|} is an H-magic labeling of G if there exist a positive ′ ′ ′

  , E integer m(f ), which we call magic sum such that for each subgraph H = (V (G) (G) ) of G ∑ ∑

  ′ f f isomorphic H, f (H ) = ′ (v) + ′ (e) = m(f ). In this case we say that the v ∈V (G) e ∈E(G)

  graph G is H-magic. When f (v) = {1, 2, ..., |V (G)|}, then G is H-supermagic and we denote supermagic-sum is s(f ).

  In [3], they proved that a complete bipartite graph K n,n could be covered by magic star covering K . Then Llad´o and Moragas [5] proved in [3] the same graph containing a cycle 1,n cover, they also proved that C -supermagic labelings on a wheel graph W n for n ≥ 5 odd and

3 C

  • supermagic labeling of a prism graph and a book graph. Marbun and Salman [6] then proved

  4 k that W n -supermagic labelings for a wheel W n -multilevel corona with a cycle C n .

  In this paper, we study an H-supermagic labeling of edge corona product of cycle and path

  • graph. We prove that a edge corona product of cycle C n and path P n graph has a P ⋄ P n

  2 supermagic labeling for n ≥ 3.

2. Main Result

  A multiset is a set that allows the existence of same elements in it(Maryati et al. [7]). Let X be a set containing some positive integers. We use the notation [a, b] to mean {x ∈ N|a ≤ x ≤ b} ∑ x and ΣX to mean . For any k ∈ N, the notation k + [a, b] means k + x|x ∈ [a, b]. x ∈X

  According to Guit´errez and Llado [3], the set X is k-equipartion if there exist k subsets of ∪ k

  |X| X , X , . . . , X

  X . say X k such that i = X and |X i | = for every i ∈ [1, k].

  1

  2 i =1 k 2.1. k-balanced multiset In this research, we used technique k-balancemultiset that introduced by Maryati et al. [7]. Let Y be a multiset of positive integers and k ∈ N. A multiset Y is k-balanced if there are k subsets of

  ∑ Y

  Y where Y = Y = Y = ... = Y then for each i ∈ [1, k]. We obtain |Y | = |Y |, ∑ Y = ∈ N i

  1 2 k i k i k ⊎ k

  Y and i = Y . i

  =1 Lemma 2.1

  [8] Let x, y, and k be integers, such that 1 ≤ x ≤ y and k > 1. If X = [x, y] and |X| is a multiple 2k, then X is k-balanced. Here, we have several lemmas on k-balanced multiset to build theorem. Lemma 2.2 Let k and x be positive integers k ≥ 3. Let Y = [1, k] ⊎ [1, k] ⊎ [x + 1, x + k], then Y is k-balanced.

  , b , c Proof. For every i ∈ [1, k] we define the multisets Y i = {a i i i } with

  ⌋ ⌊ i + 1 a i = for i ∈ [1, k]

  2 ⌉

  ⌈ i + k b = for i ∈ [1, k] i

  2 c i i = x + k + 1 − i for ∈ [1, k].

  Then, defined set A = {a |1 ≤ i ≤ k} = [1, k] i

  B = {b |1 ≤ i ≤ k} = [1, k]

i C = {c |1 ≤ i ≤ k} = [x + 1, x + k].

  i ⊎ k

  3k+3 Since A ⊎ B ⊎ C = Y and Y = Y , |Y | = 3 and ∑ Y = x + for every i ∈ [1, k], so i i i i

  =1

  2 we have Y is k-balanced.

  2

  2 Lemma 2.3 Let k and x be positive integers k ≥ 3. If Z = [x + 1, x + k ] and |Z| is k , then i Proof. For every i ∈ [1, k] we define the multisets Z i = {a |1 ≤ j ≤ k} where j

   x + i, for i ∈ [1, k] and j = 1;  i i a + 1, for j + i = k + 2; a = j j

  −1 i  a i and j others.

  • x + 1, for

  j −1 ⊎ k k 2 +1

  Z ∑ Z Since |Z i | = k; i = Z and i = (x + k ) for every i ∈ [1, k] then Z is k-balanced. i

  =1

  2 Lemma 2.4 Let x, y and k be positive integers k ≥ 4. If W = [1, x] ⊎ [1, x] ⊎ [x + 1, x + k] ⊎ [y + 1, y + k], then W is k-balanced.

  , b , c , d Proof. For every i ∈ [1, k] we define the multisets W i = {a i i i i } with a = i for i ∈ [1, k] i

  { 1 + i, for i ∈ [1, k − 1]; b i =

  1, for i = k; { x + k − i, for i ∈ [1, k − 1]; c i = x + k, for i = k; d i i = y + k + 1 − i for ∈ [1, k]

  Then, defined set A = {a i |1 ≤ i ≤ k} = [1, k] B = {b i |1 ≤ i ≤ k} = [1, k] C

  = {c i |1 ≤ i ≤ k} = [x + 1, x + k] D = {d i |1 ≤ i ≤ k} = [y + 1, y + k].

  ⊎ k W ∑ W

  Since A ⊎ B ⊎ C ⊎ D = W and i = W , |W i | = 4 and i = 5k + 2 for every i ∈ [1, k] i

  =1 then W is k-balanced.

  • Supermagic Labeling on A Cycle Graph Edge Corona with Path C

  2.2. P 2 ⋄ P n n ⋄ P n

  The edge corona product between C n and P n , denoted by C n ⋄ P n is a graph obtained by taking one copy of C n and |E(C n )| copies of P n and then joining two end-vertices of the i-th edge of C to every vertex in the i-th copy of P . n n

  Figure 1. A Cycle Graph Edge Corona with Path C n ⋄ P n Theorem 2.1 Let n be positive integers with n ≥ 3. A graph C n ⋄ P n is P ⋄ P n -supermagic.

  2

  2 Proof. Let G be a C n ⋄P n graph for any integer n ≥ 3. Then |V (G)| = n(n+1) and |E(G)| = 3n .

  2

  2 Let A = [1, 4n + n]. We define a bijective function f : V (G) ∪ E(G) → {1, 2, ..., 4n + n}.

  Here we have two cases to be considered. i Case 1. For n odd.

  Let V (G) = {v i ; 0 ≤ i ≤ n} ⊎ {u ; 0 ≤ i ≤ n, 0 ≤ j ≤ n} and j i

  E v , v v , . . . v v (G) = {v n } ⊎ {e ; 0 ≤ i ≤ n, 0 ≤ j ≤ n}. Given a set of labels for all

  1

  1 2 j

  2 vertices and edges of G denoted by A where A = [1, 4n + n]. Partition A into 3 sets,

  2

  2

  2 A = X ⊎ Y ⊎ Z, where X = [1, n] ⊎ [1, n] ⊎ [n + n + 1, n + 2n] , Y = [n + 1, n + n],

  2

  2 and Z = [n + 2n + 1, 4(n ) + n]. Then we define the total labeling f on G as follows.

  2

  2 First, partition X into 2 sets: X = [1, n] and X = [n + n + 1, n + 2n]. The vertices v i

  1

  2 where 0 ≤ i ≤ n are labeled by set X and edges {v v , v v , . . . v n v } are labeled by set

  1

  1

  1

  2

2 X . According to Lemma 2.2, if x = n + n and k = n we have n-balanced. Let X ⊎ X ,

  2 2

  1

  2 n 2n +5n+3 i then ⊎ X i = X and we have ∑ X i = . The vertices u where 0 ≤ i ≤ n and i =1 j

  2 i 0 ≤ j ≤ n are labeled by set Y . Define that u are vertices on path. According Lemma j 3 2 n

  2 +2n +n ∑ Y

  2.3, if x = n, k = n, and |Y | = n we have n-balanced where i = . Then,

  2 i i the edges e are labeled by set Z where e are edges on path and edge on product edge j j

  2

  2

  2 coronation. According Lemma 2.1, if x = n + 2n + 1, y = 4n + n, and |Z| = 3n − n we 3 2

  15n +4n −1 have n-balanced where ∑ Y i = .

2 Case 2. For n even.

  Let V (G) = {v i ; 0 ≤ i ≤ n} ⊎ {u i ; 0 ≤ i ≤ n} and E(G) = i

  {e ; 0 ≤ i ≤ n, 0 ≤ j ≤ n}. Partition A into 3 sets, A = P ⊎ Q ⊎ R, where j

  2

  2

  2 P = [1, n]⊎[1, n]⊎[n+1, 2n]⊎[2n+1, 3n], Q = [3n+1, n +n)], dan R = [n +n+1, 4n +n].

  Then we define the total labeling f on G as follows. First, partition P into 3 sets: P = [1, n],

  1 P = [n + 1, 2n], and P = [2n + 1, 3n]. The vertices v where 0 ≤ i ≤ n are labeled by

  2 3 i set P , P , and P . According to Lemma 2.4, if x = n and k = n we have n-balanced. Let

  1

  2

  3 ⊎ n

  P ⊎ P ⊎ P , then P = P and we have ∑ P = 5n + 2. The vertices u where 0 ≤ i ≤ n

  1

  2 3 i i i i =1 are labeled by set Q. Define that u i are two vertices in path. According Lemma 2.1, if

  2

  2 x = 3n + 1, y = n + n, and |Q| = 2n we have n-balanced where ∑ Q i = n + 4n + 1. Then, i i the edges e are labeled by set R where e are edges on graph G. According Lemma 2.1, if j j 3 2

  2

  2 2 15n +6n +3n x

  ∑ R = n + n + 1, y = 4n + n, and |R| = 3n we have n-balanced where i = .

  2 Furthermore, the constant supermagic sum of a subgraph P ⋄ P n are as follows

  2 {

  3

  2 8n + 4n + 3n + 1, for n odd; 3 2 f (P ⋄ P ) =

  2 n 15n +8n +21n+6

  , for n even.

  2 ⊔ ⊓

  Figure 2 illustrates an example of P 2 ⋄ P 3 -supermagic labeling on C 3 ⋄ P 3 graph.

  Figure 2.

  A P ⋄ P -supermagic labeling on C ⋄ P graph

  2

  3

  3

  3

3. Conclusion

  In this paper we have shown the P 2 ⋄ P n -supermagic labeling of edge corona product of cycle and path graph. Open Problem : For further research we can studied P ⋄ P -supermagic labeling on C ⋄ P

  2 n n m with n ≥ 3 and m ≥ 2.

  Acknowledgments We gratefully ackowledge the support from Mathematics Department of Mathematics and Natural Sciences Faculty, Universitas Sebelas Maret.

  References

  nd ed.

  [1] Chartrand, G. and L. Lesniak, Graphs and Digraphs, 2 , Wadsworth Inc., California, 1979.

[2] Gallian, J.A., A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics 19 (2016),

#DS6.

  [3] Guit´errez and A. Llad´ o, Magic Coverings, J. Combin. Math. Combin. Computing 55 (2005), 43–56.

[4] Hou, Y. and Shiu W, The Spectrum of The Edge Corona of Two Graphs, Electronic Journal of Linear Algebra

  20 (2010), 586–594. [5] Llad´ o, A. and J. Moragas, Cycle-magic Graphs, Discrete Mathematics 307 (2008), 2925–2933.

[6] Marbun, H. T. and A. N. M. Salman, wheel-Supermagic Labelings for A Wheel k-Multilevel Corona With A

Cycle

  , J. Graphs Cpmb. 10 (2013), 183–194.

[7] Maryati, T. K., A. N. M. Salman, E. T. Baskoro, J. Ryan, and M. Miller, On H-supermagic Labelings for

Certain Shackles and Amalgamations of a Connected Graph

  , Utilitas Mathematica 83 (2010), 333–342. h -supermagic Labelings of Some Trees

[8] Maryati, T. K., E. T. Baskoro, and A. N. M. Salman, P , J. Combin.

  Math. Combin. Computing 65 (2008), 182–189. [9] Sedl´ aˇck, J., Theory of Graphs and Its Applications, House Czechoslovak Sci. Prague (1964), 163–164.

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