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• W2153431281 abstract "Even vertex gracefulness of path, circuit, star and wheel are obtained. Also even vertex gracefulness of the connected graphs Cn F(2nC3), Cn F(3nC3) and C(4, n) are got. INTRODUCTION A.Solairaju, and A.Sasikala [2008] got gracefulness of a spanning tree of the graph of product of Pm and Cn, A.Solairaju and K.Chitra [2009] obtained edge-odd graceful labeling of some graphs related to paths. A.Solairaju, and C. Vimala [2008] also got the gracefulness of a spanning tree of the graph of Cartesian product of Sm and Sn . A.Solairaju and P.Muruganantham [2009] proved that ladder P2 x Pn is even-edge graceful (even vertex graceful). They found [2010] the connected graphs Pn o nC3 and Pn o nC7 are both even vertex graceful, where n is any positive integer. They also obtained [2010] that the connected graph Pn Δ nC4 is even vertex graceful, where n is any even positive integer. Section I: Preliminaries Definition 1.1: Let G = (V,E) be a simple graph with p vertices and q edges. A map f : V(G) {0,1,2,...,q} is called a graceful labeling if (i) f is one – to – one (ii) The edges receive all the labels (numbers) from 1 to q where the label of an edge is the absolute value of the difference between the vertex labels at its ends. A graph having a graceful labeling is called a graceful graph. Definition 1.2: A graph is if there exists an injective map f : E (G) {1, 2 ,..., 2q} so that the induced map f: V(G) {0, 2, 4,..., 2k-2} defined by f (x) = f(xy) (mod 2k) where k = max { p, q } makes all distinct. Definition 1.3: Cn is a circuit with n vertices. Sn is a star with n vertices. Wn is a wheel with n vertices. Section II: Even vertex graceful of standard graphs The following result is first started. Theorem 2.1: A path with n vertices is even vertex graceful. Proof: A path Pn is a connected graph with n vertices. It has (n-1) edges as follows: Some arbitrary labeling of edges of the path Pn is given below: Define f: E(Pn) → { 1,2, ..., (n-1)} by f(ei) = 2i, i varies from 1 to (n-1). Then the induced map f (u) = ∑ f(uv) (mod 2q) where the sum runs over all edges uv through v. Now, f and f+ both satisfy even vertex graceful labeling. The path Pn with n vertices is even vertex graceful. Example 2.1: The path P13 is even vertex graceful. Theorem 2.2: A circuit Cn with n vertices is even vertex graceful. Proof: Some arbitrary labeling of edges of Cn is as follows: Define f: E(Cn) → { 1,2, ..., n} by f(ei) = 2i, i varies from 1 to n. Then the induced map f (u) = ∑ f(uv) (mod 2q) where the sum runs over all edges uv through v. Now, f and f+ both satisfy even vertex graceful labeling. The path Cn with n vertices is even vertex graceful. Example 2.2: The path C11 is even vertex graceful. Theorem 2.3: A star with n vertices (Sn) is even vertex graceful. Proof: Some arbitrary labeling of edges of Sn is as follows: International Journal of Computer Applications (0975 – 8887) Volume 10– No.6, November 2010 6 Define f: E(Sn) → { 1,2, ..., n} by f(ei) = 2i, i varies from 1 to n. Then the induced map f (u) = ∑ f(uv) (mod 2q) where the sum runs over all edges uv through v. Now, f and f+ both satisfy even vertex graceful labeling. The path Sn with n vertices is even vertex graceful. Example 2.3: The star S6 is even vertex graceful. Theorem 2.4: A star with n vertices (Wn) is even vertex graceful. Proof: Some arbitrary labeling of edges of Wn is as follows: Define f: E(Cn) → { 1,2, ..., n} by f(Tj) = (2j – 1), i varies from 1 to n; n is even : f(ei) = 2q – 2(i-1), i varies from 1 to n. n is odd and n 3 (mod 4); f(en-i) = 2i +2 , i varies from 1 to (n1); f(en) = f(e1) + 2. n is odd and n 1 (mod 4); f(en-i) = 2i, i varies from 1 to (n-1); f(en) = f(e1) + 2. Then the induced map f + (u) = ∑ f(uv) (mod 2q) where the sum runs over all edges uv through v. So f and f+ both satisfy even vertex graceful labeling. The path Sn with n vertices is even vertex graceful. Example 2.4: The path W14 is even vertex graceful. Section 3 Even vertex graceful of extensionfriendship graph Definition 3.1: A fan graph or an extension-friendship graph Cn F(2nC3) is defined as the following connected graph such that every vertex of Cn is merged with one copy of 2C3. Theorem 3.1: The connected graph Cn F(2nC3) is even vertex graceful. Proof: The graph Cn F(2nC3) is chosen with some arbitrary labeling of edges as in definition (1.5). Define a map f: E[Cn F(2nC3)] {0, 1,2,..., 2q} by f(ei) = 2i-1, i = 1,2,...2n f(ti) = (2i) + f(e2n) i = 1,2,...2n f (u1) = 2q f (ui) = 2q-4i+4 i = 2,...n f (c1) = f(un) 2 f (c2) = f(c1) 6 f (ci) = f(c2) 4(i-2) i = 3,4,...n Then the induced map f (u) = ∑ f(uv) (mod 2q) where the sum runs over all edges uv through v. Now, f and f both satisfy even vertex graceful labeling as well as edge–odd graceful labeling. Thus the connected graph Cn F(2nC3 is both even vertex graceful and odd-edge graceful. Definition 3.2: A fan graph or an extension-friendship graph Cn F(3nC3) is defined as the following connected graph such that every vertex of Cn is merged with one copy of 3C3. International Journal of Computer Applications (0975 – 8887) Volume 10– No.6, November 2010 7 Theorem 3.2: The connected graph Cn F(3nC3) is even vertex graceful. Proof: The graph Cn F(3nC3) is chosen with some arbitrary labeling of edges as in definition (1.6). Define a map f: E[Cn F(3nC3)] {0, 1,2,..., 2q} by f (ei) = 2i-1, i=1,2,...,3n f (ti) = f(e2n) + 2i, i=1,2,...,2n f (u1) = (2q-4) f (u2) = (2q-6) f (ui) = f(u1)4(i-1); i=3,5,7,...,2n-1 f (ui) = f(u2)4(i-2), i=4,6,...2n f (c1) = f(u2n) – 2; f (c2) = f(c1) – 2; f (ci) = f(c1) 3(i-1) where i varies 3,5,7,...,n if n is odd; i varies 3,5,7,...,n-1 if n is even. f (cn) = f(cn-1) 4 if n is even; f (ci) = f(c2)3(i-2) where i varies 2,4,6,...,n if n is even; i varies 2,4,6,...,n-1 if n is odd; f (cn) = f(cn-1) 4 if n is odd Then the induced map f (u) = ∑ f(uv) (mod 2q) where the sum runs over all edges uv through v. Now, f and f+ both satisfy even vertex graceful labeling Thus the connected graph Cn F(3nC3 ) is even vertex graceful Definition 3.3: The graph C(4, n) is a connected graph defined by merging C4 and Cn as follows: Theorem 3.3: The connected C(4, n) is even vertex graceful. Proof: The graph Cn F(3nC3) is chosen with some arbitrary labeling of edges as in definition (1.7). Define a map f: E [C(4, n)] {0, 1,2,..., 2q} by f (ei) = 2i-1, i=1,2,..., (n+3); f (Ti) = 2q – 2(i-1), i=1,2,..., (n+3); f (e0) = (q -15) = (2n-8) Then the induced map f (u) = ∑ f(uv) (mod 2q) where the sum runs over all edges uv through v. Now, f and f+ both satisfy even vertex graceful labeling Thus the connected graph C(4, n) is even vertex graceful. CONCLUSION Even vertex graceful of friendship graphs F(nC3), F(nC5), and F(2nC3) are obtained in [7]. For further investigations. Path, circuit, star and wheel are all even vertex graceful. Also the connected graphs Cn F(2nC3), Cn F(3nC3) and C(4, n) are all even vertex graceful.." @default.
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• W2153431281 title "Even Vertex Graceful of Path, Circuit, Star, Wheel, some Extension-friendship Graphs and Helm Graph" @default.
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