# Enhanced Algorithms for VMTL, EMTL and TML on Cycles and Wheels

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 249)

## Abstract

This paper deals with the labeling of vertices and edges of a graph. Let G be a graph with vertex set V and edge set E, where |V| be the number of vertices and |E| edges of G. The two bijection methods for which we have designed algorithms are as follows. Initially A bijection λ 1:V∪E → {1, 2…|V| + |E|} is called a Vertex-Magic Total Labeling (VMTL) if there is a vertex magic constant vk such that the weight of vertex m is, λ 1(m) + Σ n εA (m) λ 1(mn) = vk, ∀ m εv Where A(m) is the set of vertices adjacent to x. In the similar fashion the bijection λ 2:V∪E → {1, 2…|V| + |E|} is called Edge-Magic Total Labeling (EMTL) if there is a edge magic constant ek such that the weight of an edge es(mn), λ 2(m) + λ 2(n) + λ 2 e(mn) = ek, ∀ e εE. A resultant Graph which consists of both VMTL and EMTL are said to be Total Magic Labeling (TML) for different vertex magic constant and edge magic constant values. Baker and Sawada proposed algorithms to find VMTLs on cycles and wheels. In this paper we enhanced these algorithms and also we proposed new algorithms to generate EMTLs and TMLs of cycles and wheels. We used the concept variations and sum set sequences to produce VMTLs and EMTLs on cycles and wheels. Also we designed modules to identify TML’s.

### Keywords

Magic labeling Magic labeling algorithms Magic constant Cycles Wheels Vertex Magic Total Labeling Edge Magic Total Labeling Total Magic Labeling etc

## Preview

### References

1. 1.
Baker, A., Sawada, J.: Magic labelings on cycles and wheels. In: Yang, B., Du, D.-Z., Wang, C.A. (eds.) COCOA 2008. LNCS, vol. 5165, pp. 361–373. Springer, Heidelberg (2008)
2. 2.
Gallion, J.A.: A dynamic survey of graph labeling. Electronic J. Combinations 14, DS6 (2007)Google Scholar
3. 3.
West, D.B.: An introduction to graph theory. Prentice-Hall (2004)Google Scholar
4. 4.
Gray, I., McDougall, J., Wallis, W.D.: On Vertex labeling of complete Graphs. Bull., Inst., Combin., Appl. 38, 42–44 (2003)
5. 5.
MacDougall, J.A., Miller, M., Slamin, Wallis, W.D.: Vertex Magic Total Labelings of Graphs. Utilitas Math. 61, 3–21 (2002)
6. 6.
MacDougall, J.A., Miller, M., Wallis, W.D.: Vertex-magic total labelings of wheels and related graphs. Utilitas Mathematica 62, 175–183 (2002)
7. 7.
Andersen, H.R., Hulgaard, H.: Boolean Expression Diagrams. Information and Computation 179, 194–212 (2002)