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

#### Abstract

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 *G*_{1} and *G*_{2}, denoted by *G*_{1}[*G*_{2}], is a graph which arises from *G*_{1} by replacing each vertex of *G*_{1} by a copy of the *G*_{2} and each edge of *G*_{1} by all edges of the complete bipartite graph *K*_{n, n} where *n* is the order of *G*_{2}. In this paper we provide a sufficient condition for $\overline{C_{n}}[\overline{K_{m}}]$ in order to have a $P_{t}[\overline{K_{m}}]$-magic decompositions, where *n* > 3, *m* > 1, and *t* = 3, 4, *n* − 2.

#### Keywords

#### Full Text:

PDF#### References

D. Froncek, P. Kovar, T. Kovarova, Constructing Distance Magic Graphs From Regular Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing. 78 (2011),

–354

M. Baca, M. Miller, Super edge-antimagic graphs, Brown Walker Press, Boca Raton, Florida USA (2008).

H. Enomoto, A. Llado, T. Nakamigawa, G. Ringel, Super edge magic graphs, SUT Journal of Mathematics. 34 (1998), 105–109.

J. A Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics. ]DS6, 2016.

Gutierrez, A. Llado, A Magic Coverings, Journal of Combinatorial Mathematics and Combinatorial Computing. 55 (2005) 43–56

Hendy, The H-super (anti) magic Decomposition of Antiprism graphs, AIP Conference Proceedings 1707. 020007(2016);DOI: 10.1063/1.4940808.

Hendy, An H-super magic Decompositions of The Lexicographic Product of Graphs, preprint.

Inayah, A. Llado, J. Moragas, Magic and Antimagic H decompositions, Discrete Math. 312 (2012) 1367–1371.

A. Kotzig, A. Rosa, Magic valuation of finite graphs, Canadian Mathematics Bulletin. 13, (1970) 451–461.

Z. Liang, Cycle-supermagic decompositions of Complete multipartite Graphs, Discrete Mathematics. 312, (2012) 3342–3348.

T.K. Maryati, A.N.M. Salman, On graph-(super)magic labelings of a path-amalgamation of isomorphic graphs, Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications. (2010) 228–233.

K.A. Sugeng, Magic and Antimagic labeling of graphs, University of Ballarat,(2005).

W.D. Wallis, Magic Graphs, Birkhauser Boston, Basel, Berlin (2001).

DOI: http://dx.doi.org/10.19184/ijc.2018.2.2.2

### Refbacks

- There are currently no refbacks.

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.