TOTALLY MAGIC CORDIAL LABELING OF SOME SPECIAL GRAPHS Dr. P. Lawrence Rozario Raj Assistant Professor, Department of Mathematics, St. Joseph’s College Trichirappalli – 620 002, Tamil Nadu, India. lawraj2006@yahoo.co.in
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All graphs in this paper are finite, simple and undirected All graphs in this paper are finite, simple and undirected. We follow the basic notation and terminology of graph theory as in Harary and of graph labelling as in Gallian. Cahit defines cordial labeling and cordial graph. The concept of totally magic cordial labeling is introduced by Cahit. He proves that the following graphs have a TMC labeling: Km,n (m,n>1), trees, cordial graphs, and Kn if and only if n = 2,3,5, or 6.
Definition 1.1 The assignment of values subject to certain conditions to the vertices of a graph is known as graph labeling. Definition 1.2 Let G = (V, E) be a graph. A mapping f : V(G) →{0,1} is called binary vertex labeling of G and f(v) is called the label of the vertex v of G under f. For an edge e = uv, the induced edge labeling f*:E(G)→{0,1} is given by f*(e) = |f(u) − f(v)|. Let vf(0), vf(1) be the number of vertices of G having labels 0 and 1 respectively under f and let ef(0), ef(1) be the number of edges having labels 0 and 1 respectively under f*. Definition 1.3 A binary vertex labeling of a graph G is called a cordial labeling if | vf(0) − vf(1) | 1 and | ef(0) − ef(1) | 1. A graph G is cordial if it admits cordial labeling.
Definition 1.4 [3] A graph G(p,q) is said to have a totally magic cordial (TMC) labeling with constant C if there exists a mapping f : V(G)E(G){0,1} such that f(a)+f(b)+f({a,b}) = C (mod 2) for all {a,b}E(G) provided the condition |f(0)–f(1)|1 is hold, where f(0) = vf(0) + ef(0) and f(1) = vf(1) + ef(1) and vf(i), ef(i); i {0,1} are, respectively, the number of vertices and edges labeled with i. Definition 1.5 [4] If G has order n, the corona of G with H, GH is the graph obtained by taking one copy of G and n copies of H and joining the ith vertex of G with an edge to every vertex in the ith copy of H. Definition 1.6 If G has order n, the k-corona of G with H, GkH is the graph obtained by taking k copies of G and n copies of H and joining the ith vertex of every copies of G with an edge to every vertex in the ith copy of H.
2. Totally magic cordial labeling of planar graphs Pln and Plm,n In this section, we show that two classes of planar graphs whose definitions are based on complete graphs and complete bipartite graphs are shown to be totally magic cordial.
3. Totally magic cordial labeling of S(Pn), S(Cn) and S(Km,n) In this section, we show that the splitting graph of path, cycle and complete bipartite graph are totally magic cordial. Definition 3.1 [7] For each vertex v of a graph G, take a new vertex v. Join v to all the vertices of G adjacent to v. The graph S(G) thus obtained is called splitting graph of G.
4. Totally magic cordial labeling of 2-corona of Pn and Nm In this section, we show that the 2-corona of Pn and Nm is totally magic cordial.
5. Conclusion Labeled graph is the topic of current interest for many researchers as it has diversified applications. I discuss here totally magic cordial labeling of some special graphs. This approach of these graphs is novel and contributes to new graphs to the theory of totally magic cordial graphs. The results reported here are new and will add new dimension in the theory of totally magic cordial graphs.
References Baskar Babujee J, Planar graphs with maximum edges anti-magic property, The Mathematics Education, 37(4), (2003) 194-198. Cahit I, Cordial Graphs : A weaker version of graceful and harmonious Graphs, Ars Combinatoria, 23, (1987) 201-207. Cahit I, Some totally modular cordial graphs, Discuss. Math. Graph Theory, 22, (2002) 247-258. Frucht R and Harary F, On the corona of two graphs, Aequationes Math, 4, (1970) 322-325. Gallian J A, A dynamic survey of graph labeling, The Electronics Journal of Combinatorics, 16, (2009) DS6. Harary F, Graph theory, Addison Wesley, Reading, Massachusetts, 1972. Sampathkumar E and Walikar H B, On splitting graph of a graph, J. Karnatak Univ. Sci., 25 and 26 (Combined) (1980-81), 13-16. Ramanjaneyulu K, Venkaiah V Ch, and Kishore Kothapalli, Anti-magic labelings of a class of planar graphs, Australasian Journal of Combinatorics, 41, (2008) 283-290.
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