Totally Magic Cordial Labeling of Some Special Graphs P. Lawrence Rozario Raj 1 and R. Lawrence Joseph Manoharan 2 1 Assistant Professor, Department of Mathematics, St. Joseph’s College Trichirappalli–620 002, Tamil Nadu, India 2 Associate Professor and Head (Rtd.), Department of Mathematics, St. Joseph’s College, Tiruchirappalli–620 002, Tamil Nadu, India E-mail: lawraj2006@yahoo.co.in, rljmano@gmail.com Abstract—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) = v f (0) + e f (0) and f(1) = v f (1) + e f (1) and v f (i), e f (i); i ∈ {0,1} are, respectively, the number of vertices and edges labeled with i. In this paper, we present the totally magic cordial (TMC) labeling of two families of planar graphs, Pl n and Pl m,n , splitting graph of path, cycle and complete bipartite graph and some special corona graphs. Keywords: Cordial labeling, Magic graphs, Totally magic cordial labeling, Totally magic cordial graph. 2010 AMS Subject Classification: 05C78 I NTRODUCTI ON All graphs in this paper are finite, simple and undirected. We follow the basic notation and terminology of graph theory as in Harary [6] and of graph labelling as in [5]. In [2], Cahit defines cordial labeling, a variation of both graceful and harmonious labelings. The concept of totally magic cordial labeling is introduced by Cahit [3]. He proves that the following graphs have a TMC labeling: K m,n (m,n>1), trees, cordial graphs, and K n 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 v f (0), v f (1) be the number of vertices of G having labels 0 and 1 respectively under f and let e f (0), e f (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 | v f (0) − v f (1) | ≤ 1 and | e f (0) − e f (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) = v f (0) + e f (0) and f(1) = v f (1) + e f (1) and v f (i), e f (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, G☼H is the graph obtained by taking one copy of G and n copies of H and joining the i th vertex of G with an edge to every vertex in the i th copy of H. Definition 1.6 : If G has order n, the k-corona of G with H, G k ☼H is the graph obtained by taking k copies of G and n copies of H and joining the i th vertex of every copies of G with an edge to every vertex in the i th copy of H. 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. We first define the graphs below.