On
P
2⋄
P
n-Supermagic Labeling of Edge Corona
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 graphG= (V, E) admits aH-covering, whereHis subgraph ofG, if every edge inE belongs to a subgraph ofGisomorphic toH. Graph GisH-magic if there is a total labelingf:V(G)∪E(G)→1,2, ...,|V(G)|+|E(G)|, such that each subgraphH′= (V′, E′) of Gisomorphic toH and satisfyingf(H′)def
= ΣvϵV′f(v) + ΣeϵE′f(e) =m(f) wherem(f) is a
con-stant magic sum. Additionaly,GadmitsH-supermagic iff(V) = 1,2, ...,|V|. The edge corona
Cn⋄PnofCnandPnis defined as the graph obtained by taking one copy ofCnandncopies of
Pn, and then joining two end-vertices of thei-th edge ofCnto every vertex in thei-th copy ofPn. This research aim is to findH-supermagic covering on an edge corona product of cycle and path graphCn⋄PnwhereHisP2⋄Pn. We usek-balanced multiset to solve our reserarch. Here, we
find that an edge corona product of cycle and path graphCn⋄PnisP2⋄Pnsupermagic forn≥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 nvertices is denoted byCn. They also defined
path graph is a walk with no repeated vertices, path graph with nvertices is denoted byPn.
Let G1 and G2 are two graphs on disjoint sets of n1 and n2 vertices, m1 and m2 edges,
respectively. The edge corona G1⋄G2 is defined as the graph obtained by taking one copy of G1 andm1copies ofG2, and then joining two end-vertices of thei-th edge ofG1 to every vertex
in thei-th copy ofG2. Note that the edge corona G1⋄G2 of G1 andG2 hasn1+m1n2 vertices
and m1+ 2m1n2+m1m2 edges, for detail defnition of graph see [4].
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 H1, H2, ..., Hk such that each edge of E belongs to at least
one of the subgraphs Hi, 1 ≤ i ≤ k. Then, it is said that G admits an (H1, H2, ..., Hk
)-edge covering. If every Hi 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 anH-magic labeling ofGif there exist a positive integer m(f), which we call magic sum such that for each subgraph H′ = (V(G)′, E(G)′) of G isomorphic H, f(H′) = ∑
v∈V(G)′f(v) +
∑
graph G is H-magic. When f(v) = {1,2, ...,|V(G)|}, then G is H-supermagic and we denote supermagic-sum iss(f).
In [3], they proved that a complete bipartite graph Kn,n could be covered by magic star
covering K1,n. Then Llad´o and Moragas [5] proved in [3] the same graph containing a cycle
cover, they also proved that C3-supermagic labelings on a wheel graph Wn for n≥5 odd and
C4-supermagic labeling of a prism graph and a book graph. Marbun and Salman [6] then proved that Wn-supermagic labelings for a wheelWnk-multilevel corona with a cycleCn.
In this paper, we study anH-supermagic labeling of edge corona product of cycle and path graph. We prove that a edge corona product of cycle Cn and path Pn graph has a P2⋄Pn
-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]). LetX be a set containing some positive integers. We use the notation [a, b] to mean {x∈N|a≤x≤b}
and ΣX to mean ∑
x∈Xx. For any k ∈ N, the notation k+ [a, b] means k+x|x∈[a, b].
According to Guit´errez and Llado [3], the set X is k-equipartion if there exist k subsets of X. sayX1, X2, . . . , Xk such that∪ki=1Xi =X and|Xi|= |Xk| for every 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 andk∈N. A multisetY isk-balanced if there areksubsets of Y whereYi =Y1=Y2 =...=Ykthen for eachi∈[1, k]. We obtain|Yi|=|Yk|,
Here, we have several lemmas onk-balanced multiset to build theorem.
Proof. For everyi∈[1, k] we define the multisetsZi ={aij|1≤j≤k}where
aij =
x+i, for i∈[1, k]and j = 1; ai
j−1+ 1, for j+i=k+ 2;
ai
j−1+x+ 1, for i and j others.
Since|Zi|=k;⊎ki=1Zi =Z and∑Zi= (x+k2)k+12 for everyi∈[1, k] thenZ isk-balanced.
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 isk-balanced.
Proof. For everyi∈[1, k] we define the multisetsWi={ai, bi, ci, di} with
ai = i for i∈[1, k]
bi = {
1 +i, for i∈[1, k−1];
1, for i=k;
ci = {
x+k−i, for i∈[1, k−1]; x+k, for i=k; di = y+k+ 1−i for i∈[1, k]
Then, defined set
A = {ai|1≤i≤k}= [1, k]
B = {bi|1≤i≤k}= [1, k]
C = {ci|1≤i≤k}= [x+ 1, x+k]
D = {di|1≤i≤k}= [y+ 1, y+k].
SinceA⊎B⊎C⊎D=W and⊎k
i=1Wi =W,|Wi|= 4 and∑Wi = 5k+ 2 for everyi∈[1, k]
then W isk-balanced.
2.2. P2⋄Pn-Supermagic Labeling on A Cycle Graph Edge Corona with Path Cn⋄Pn
The edge corona product betweenCnandPn, denoted byCn⋄Pn is a graph obtained by taking
one copy of Cn and |E(Cn)| copies of Pn and then joining two end-vertices of the i-th edge of
Cn to every vertex in thei-th copy ofPn.
Theorem 2.1 Let nbe positive integers with n≥3. A graph Cn⋄Pn is P2⋄Pn-supermagic.
Proof. LetGbe aCn⋄Pngraph for any integern≥3. Then|V(G)|=n(n+1) and|E(G)|= 3n2.
Let A= [1,4n2+n]. We define a bijective functionf :V(G)∪E(G)→ {1,2, ...,4n2+n}.
Here we have two cases to be considered.
Case 1. For n odd. Let V(G) = {vi; 0 ≤ i ≤ n} ⊎ {uij; 0 ≤ i ≤ n,0 ≤ j ≤ n} and
j are labeled by set Z where eij are edges on path and edge on product edge
coronation. According Lemma 2.1, if x=n2+ 2n+ 1, y = 4n2+n, and|Z|= 3n2−nwe
are labeled by set Q. Define that ui are two vertices in path. According Lemma 2.1, if
x= 3n+ 1,y =n2+n, and|Q|= 2nwe have n-balanced where∑
Furthermore, the constant supermagic sum of a subgraph P2⋄Pn are as follows
f(P2⋄Pn) =
Figure 2. A P2⋄P3-supermagic labeling on C3⋄P3 graph
3. Conclusion
In this paper we have shown the P2⋄Pn-supermagic labeling of edge corona product of cycle
and path graph.
Open Problem: For further research we can studied P2⋄Pn-supermagic labeling onCn⋄Pm
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
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[5] Llad´o, A. and J. Moragas,Cycle-magic Graphs, Discrete Mathematics307(2008), 2925–2933.
[6] Marbun, H. T. and A. N. M. Salman, wheel-Supermagic Labelings for A Wheelk-Multilevel Corona With A Cycle, J. Graphs Cpmb.10(2013), 183–194.
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