International Journal of Computer Applications (0975 – 8887) Volume 10– No.10, November 2010 1 Edge-odd Gracefulness of Product of P 2 and C N Dr. A. Solairaju Associate Professor of Mathematics Jamal Mohamed College, Tiruchirapalli – 620 020. Tamil Nadu, India. A.Sasikala and C. Vimala Assistant Professors (SG), Department of Mathematics, Periyar Maniammai University, Vallam Thanjavur – Post.. Tamil Nadu, India. ABSTRACT A (p, q) connected graph is edge-odd graceful graph if there exists an injective map f: E(G) → {1, 3, …, 2q-1} so that induced map f + : V(G) → {0, 1,2, 3, …, (2k-1)}defined by f + (x)  f(x, y) (mod 2k), where the vertex x is incident with other vertex y and k = max {p, q} makes all the edges distinct and odd. In this article, the Edge- odd gracefulness of strong product of P 2 and C n is obtained. Key words: , Graceful Graphs, Edge-odd graceful labeling, Edge-odd Graceful Graph 1. INTRODUCTION A.Solairaju and K.Chitra [2009] obtained edge-odd graceful labeling of some graphs related to paths. A. Solairaju et.al. [2009, 2010] proved that the square 2-nC 4 , 3-nC 4, 4-nC 4 are edge -odd graceful. Section-2: Edge-odd graceful labeling of strong product of P 2  C n Definition 2.1: Graceful Graph: A function f of a graph G is called a graceful labeling with m edges, if f is an injection from the vertex set of G to the set {0, 1, 2, …, m} such that when each edge uv is assigned the label │f(u) – f(v)│ and the resulting edge labels are distinct. Then the graph G is graceful. Definition 2.2: Edge-odd graceful graph: A (p, q) connected graph is edge-odd graceful graph if there exists an injective map f: E(G) → {1, 3, …, 2q-1} so that induced map f+: V(G) → {0, 1, 2, …,(2k-1)} defined by f+(x)   f(x, y) (mod 2k), where the vertex x is incident with other vertex y and k = max {p, q} makes all the edges distinct and odd. Hence the graph G is edge- odd graceful. Theorem 2.1: The strong product of P 2 C n is edge-odd graceful. Proof: The strong product of the path P 2 and the circuit C n is given and the arbitrary labelings for vertices and edges for P 2 C n are mentioned below. Figure 1: Edge-odd graceful Graph of P 2  C n To find edge-odd graceful, define by f: E(P 2  C n ) → {1, 3, …, 2q- 1} Case i . n  0 (mod 5) f(e 1 ) = 5, f(e 2 ) =1, f(e 3 ) = 7, f(e 4 ) = 3 f(e i ) = 2i-1, i = 5,6,7,…5n Rule(1) Case ii. n 2(mod 5) f(e 1 ) = 5, f(e 2 ) = 3, f(e 3 ) = 1, f(e 4 ) = 7 f(e i ) = 2i-1, i = 5,6,7,…,(2n+2), (3n+3), (3n+4), …, 5n f(e 3n+3-i ) = f(e 2n+2 )+2i, i=1,2,……n Rule (2) Case iii. n  4 (mod 5) f(e 1 ) = 5, f(e 2 ) = 7, f(e 3 ) = 1, f(e 4 ) = 3 f(e i ) = 2i-1, i = 5,6,7,…,(2n+2), (3n+3), (3n+4), …, 5n f(e 3n+3-i ) = f(e 2n+2 )+2i, i=1,2,……n Rule (3) Case iv. n  3 (mod 5) f(e 1 ) = 7, f(e 2 ) = 1, f(e 3 ) = 3, f(e 4 ) = 5 f(e i ) = 2i-1, i = 5,6,7,…,(2n+2), (3n+3), (3n+4), .., 5n f(e 3n+3-i ) = f(e 2n+2 )+2i, i=1,2,……n Rule (4) For n  1 (mod 5), the arbitrary labelings for vertices and edges for P 2 C n are mentioned below e2 en+4 en+5 en+6 e2n-1 e2n+1 e2n+2 e1 e3n+5 e3n+7 e5n-2 e3n+2 e3n+1 e3n e2n+6 e2n+5 e2n+4 e5n e2n+3 e3n+4 e3n+6 e5n-1 e3n+3 e3n+8 e5n-3 e5 e6 e7 en+1 en+2 en+3