Another H-super magic decompositions of the lexicographic product of graphs

Let H and G be two simple graphs. The concept of an H-magic decomposition of G arises from the combination between graph decomposition and graph labeling. A decomposition of a graph G into isomorphic copies of a graph H is H-magic if there is a bijection f : V (G) ∪ E(G) −→ {1, 2, ..., |V (G)∪E(G)|} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. A lexicographic product of two graphs G1 and G2, denoted by G1[G2], is a graph which arises from G1 by replacing each vertex of G1 by a copy of the G2 and each edge of G1 by all edges of the complete bipartite graph Kn,n where n is the order of G2. In this paper we provide a sufficient condition for Cn[Km] in order to have a Pt[Km]-magic decompositions, where n > 3,m > 1, and t = 3, 4, n− 2.


Introduction
Let G be a simple graph and H be a subgraph of G.A decomposition of G into isomorphic copies of H is called H− magic if there is a bijection f : V (G) ∪ E(G) −→ {1, 2, ..., |V (G) ∪ E(G)|} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant.A lexicographic product of two graphs G 1 and G 2 is defined as graph which constructed from the graph G 1 and then replacing each vertex of G 1 by a copy of G 2 and each edge of G 1 by edges of complete bipartite graph K n,n , where |V (G)| = n.The lexicographic product of G 1 and G 2 with this construction is denoted by A labeling of a graph G = (V, E) is a bijection from a set of elements of graphs to a set of numbers.The edge magic and super edge magic labelings were first introduced by Kotzig and Roza [9] and Enomoto, Lladò, Nakamigawa, and Ringel [3], respectively.There are some results in edge magic and super edge magic, such as in [2,3,12,13].The notion of an H− (super) magic labeling was introduced by Gutièrrez and Lladò [5] in 2005.In 2010, Maryati and Salman [11] used multiset partition concept to obtain a super magic labeling of path amalgamation of isomorphic graphs.Inayah et al. [8]have improved the concept of labeling graphs became H−(anti) magic decomposition.In almost the same time, Liang [10] discused cycle-supermagic decompositions of complete multipartite graphs and in 2015, Hendy [6] has discused the H− super(anti)magic decompositions of antiprism graphs.For a complete results in graph labeling, see [4].
In this research we interest in decomposing the lexicographic product of graphs C n [K m ] then labeling of the edges and vertices of each isomorphic copies of P t [K m ] to obtain P t [K m ]− magic decomposition, where n > 3, m > 1, and t = 3, 4, n − 2.

Preliminaries
Let G be a simple graph.Complement of G, denoted by G, is graph which In such case, we write G = G 1 ⊕ G 2 ⊕ ... ⊕ G t and G is said to be H-decomposable.if G is an H-decomposable graph, then we also write H|G.
Let B is an H-decomposition of G.The graph G is said to be H-magic if there exists a bijection The sum of all the vertex and edges labels of H (under a labeling f ) is denoted by f (H).The constant value that every copy of H takes under the labeling f is denoted by m(f ).
The one of the concept of multi set partition, k-balance multi set, was presented by Maryati et al. [11].In this paper, x∈X x, denoted by X. Multi set is a set which may has the same elements.For positive integer n and Theorem 1.1.[7] Let n and m be integers with n > 3 and m > 1.The graph C n [K m ] has P 2 [K m ]super magic decomposition if and only if m is even or m is odd and n ≡ 1(mod4), or m is odd and n ≡ 2(mod4), or m is odd and n ≡ 3(mod4).
then from Lemma 2.1 we have that P 3 |C n .From Lemma 2.2 we have that n ≡ 0(mod4) or n ≡ 3(mod4).Because of C 4 doesn't have P 3 , this is not occur for Algorithm 1: as a center of this rotation until v 2 on position of v 2k and we have k P 3 -path.
From the Algorithm 1 above, we have that P 3 |C n .Then from Lemma 2.1 See Figure 1 to see graph C 8 can be decomposed into 10 P 3 -path.
Let m be even.Do the next vertex labeling steps and edge labeling steps such in case 1 in Theorem 2.1.
Now let m is odd.Do the vertex labeling steps and edge labeling steps such in case 4 in Theorem 2.1.
if j is even.
and label all vertices in every V i with the elements of A i .Define a function 2 )]} B and label all edges in every P Follow the same way with (a),for m = 3, A is a (n+ n(n−3)

2
)-balance multi set and for every i ∈ , where if j is even.
if j is even.
and label all vertices in every V i with the elements of ]-magic decomposition.Now let n ≡ 0(mod4) and m be even.From Lemma 3, we have for n ≡ 0(mod4), Do the vertex labeling steps and edge labeling steps such in case 1 in Theorem 2.1.Since for all  Figure 3 shows that graph C 9 can be decompose into 9 P 4 -path.2 , (n + n(n−3)
]; 2 With the same way for m = 3, A is (n + n(n−3)

)],
A i is a balance subset of A. Consider the set E = [3(n + n(n−3) , where if j is even.
and label all vertices in every V i with the elements of Now let n ≡ 9(mod12).From Lemma 2.2 we have that for n ≡ 9(mod12), P . and let v 1 be the center of the rotation.Rotate L1 such that v 1 on v 2 , v 3 on v 4 , v n on v 1 and etc. Do the next rotation such that v 1 on v 3 ,...etc, and continue the process until all edge are used up.

Figure 2
Figure 2 give an example that graph C 8 [K 2 ] have P 3 [K 2 ]super magic decomposition with the constant value m(f 1 + f 2 ) = 503.

Figure 3 .
Figure 3. P 4 -decomposition of C 9 magic decomposition.Let m is odd.Do the vertex labeling steps and edge labeling steps such in case 3 in Theorem 2.1.Let m = 3.Consider the set D = [1, m(n + n(n−3)

2 ]
with the elements of A n+i B i .Let m > 3 and m be odd.Consider the set A * = [1, m(n + n(n−3) 2 )].Divide A * to be the two sets A and E where A = [1, 3(n + n(n−3)

6 ]
4 [K m ]|C n [K m ].Now, let m be even.Do the vertex labeling steps and edge labeling steps such in case 1 in Theorem 2.1.Because ∀i ∈ [1, n(n−3) , (f 1 + g)(P 4 [K mi ) = 4 Z i + 3 X i then C n [K m ] have P 4 [K m ]magic decomposition.Suppose m is odd.Do the vertex labeling steps and edge labeling steps such in case 2 of Theorem 2.1.Since for all magic decomposition.Now let n ≡ 0(mod12) and m be even.Clearly from Lemma 2.2 that for n ≡ 0(mod12),P 4 [K m ]|C n [K m ].Do the vertex labeling steps and edge labeling steps such in case 1 of Theorem 1.Because

Figure 4 . 2 ). 4 ) 2 ) 2 ) 2 ) + 1 + 2 ) 2 ).
Figure 4. P 9 -decomposition of C 12 let v 1 be the center of the rotation.Rotate P 1 such that v 1 on v 3 , v 3 on v 5 and v 4 on v 6 , thus we have P 2 : v 5 − v 3 − v 6 .Do the next rotation until v 1 on v 5 ,..., v 4i−1 ,..., v 4k−1 .Then we have 2k of P 3 -paths.2 Choose the cycle v 2 − v 4 − ... − v 4k .Decompose this 2k-cycle to k of P 3 -paths.3 Do the rotation again (v 1 → v 3 → v 5 →...), with choosing two vertices which close with the vertices that is rotated in step 1.If this rotation is not the last rotation, do the rotation again until v 1 on position of v 4k−1 , such that we have 2k of P 3 -path.If this rotation is the last rotation, first do the rotation in step 1 until v 1 on position of v 2k−1 such that we have k of P 3 -path.Then rotate magic decomposition.Let m be odd.Do the next vertex labeling steps and edge labeling steps such in case 4 in Theorem 2.1.Since for all