Graceful Labelings for Sun Graph, Sun Extension Graph, Double Fan, and Leg Graph with Star

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 5 (017), pp Research India Publications Graceful Labelings for Sun Graph, Sun Extension Graph, Double Fan, and Leg Graph with Star A. Solairaju 1, S. Subbulakshmi, and R. Kokila 3 1 Associate Professor of Mathematics, Jamal Mohamed College (Autonomous), Trichy, India. Assistant Professor of Mathematics, Dharmapuram Gnanabigai Government Arts College (Women), Mayiladuthurai , Tamilnadu, India. 3 Head and Assistant Professor of Mathematics, Sindhi College of Arts & Science, 146, Poonamallee High Road, Numbal, Chennai , Tamilnadu, India. Abstract Abhyankar [00] made a conclusion that the graphs obtained from Cm+1 P5 by adding a pendent edge to each vertex of an outer cycle is graceful. Sekar [00] settled that the graph Cm Pn obtained by attaching the path Pn to each vertex of Cm is graceful, and also that Cn is graceful if and only if n 0 mod 4; the graph obtained by identifying a vertex of C4k+ with an endpoint of a star are one modulo three graceful graph. Bhat-Nayak and Selvam [003] proved that the n-cone Cm + Kn is to be graceful when m 0 or 3 (mod 1). When n is even and m is, 6 or 10 (mod 1), Cm + Kn violates the parity condition for a graceful graph. They also contributed that the following cones are graceful: C4 + Kn; C5 + K; C7 + Kn; C9 + K; C11 + Kn and C19 + Kn. In this article, graceful labelings are obtained for the graphs sun, extension sun, double fan and leg graph with star are obtained. Keywords: Graceful graph, sun graph, sun extension graph, double fan, and leg graph with star.

2 1358 A. Solairaju, S. Subbulakshmi and R. Kokila INTRODUCTION Rosa [1967] got that the n-cycle Cn is graceful if and only if n 0 or 3 (mod 4). Let Cn (t) denote the one-point union of t cycles of length n. Bodendiek, Schumacher, and Wegner [1975] brought a result that the one-point union of any two cycles is graceful when the number of edges is congruent to 0 or 3 (mod 4). Bermond, Brouwer, and Germa [1978] and Bermond, Kotzig, and Turgeon [1978] proved that C3 (t) (the friendship graph or Dutch t-windmill) is graceful if and only if t 0 or 1 (mod 4). Koh, Rogers, Lee, and Toh in [1979] conjectured that Cn (t) is graceful if and only if nt 0 or 3 (mod 4). Delorme, Maheo, Thuillier, Koh, and Teo [1980] and, Ma & Feng [1984] obtained that any cycle with a chord is graceful. They also proved that all cycles with a Pkchord are graceful. Graham and Sloane [1980] proved that wheels Wn = Cn + K1 are graceful. Lee and Seah [1990] got that Cn k (the k th power of Cn) is edge- graceful for k < n/ if and only if n is odd and Cn k is edge-graceful for k n/ if and only if n (mod 4). Gallian [1989] conjectured that every graph obtained by adding a single pendent edge to one or more vertices of a cycle is graceful, and for all n and t, all members of C +t n are graceful. Ropp [1990] got that for every n and t, the class C +t n contains a graceful graph. Shee [1991] obtained that C4 (t) is graceful for all t. He analyzed the conjecture and in the case that the cycle is even, he showed that the graphs have α-labeling. Huang and Skiena [1994] determined that Cm Pn is graceful for all n when m is even and for all n with 3 n 1 when m is odd. Guo [1994] obtained that dragons are graceful when the length of the cycle is congruent to 1 or (mod 4). Liu [1995] analyzed that the graph obtained from a gear graph by attaching one or more pendant points to each vertex between the cycle vertices is graceful. For i = 1,,, n, let vi,1, vi, vi,m be the successive vertices of n copies of Cm. Sekar [00] defined a chain of cycles Cm,n as the graph obtained by identifying vi,m and vi+1,m for i = 1,,, (n 1). He found that C6,k and C8,n are graceful for all k and all n. Barrientos [00] analyzed that helms (graphs obtained from a wheel by attaching one pendent edge to each vertex) are graceful. Sethuraman and Elumalai [005] got that every cycle Cn (n 6) with parallel Pkchords is graceful for k = 3, 4, 6, 8, and 10 and they conjectured that the cycle Cn with parallel Pk-chords is graceful for all even k. SECTION 1: BASIC DEFINITIONS AND PRELIMINARIES Definition 1.1: Path: An open walk in which no vertex appears more than once is called a path. We denote Pn as a path with n vertices where n 1. Definition 1.: Circuit: A closed walk in which no vertex (except the initial and the final vertex) appears more than once is called a circuit. We denote Cn as a circuit with n vertices where n 1.

3 Graceful Labelings for Sun Graph, Sun Extension Graph, Double Fan, Definition 1.3: 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,,, m} such that when each edge uv is assigned the label f(u) f(v) the resulting edge labels are distinct. Hence the graph G is graceful. SECTION : DIFFERENT GRACEFUL LABELINGS FOR FEW GRAPHS The following theorem is first started: Theorem.1: The sun graph Pn+1 np is graceful. Proof: The given connected graph G = Pn+1 np has the vertex set = {V0,V1,,Vn}, and edge set is {ViVi+1: i = 1,,, (n-1), (n+1),, (n-1)} {VnV0,V0Vn+1} {ViVn+-i : i =1 to n}. One of the arbitrary labelings of the graph G for vertices is as follows (figure 1): To get graceful labeling, define map f : V(G) { 0,1,,,q} by f(v n+1 i ) = q ( i 1 ) ; i = 1,3,5 n 1; = i ; i =,4,,n. f(v n+1 ) = 0; f (V0) = n+1 (n is odd); = q - n (n is even). The induced map f + : E(G) {1,, 3,, q} from the map f itself is defined by f+ (uv) = f(u) f(v) for each edge uv in E(G) of the given graph. Here f + is an injective map. Both f and f + satisfy the conditions for graceful labeling. Hence the sun graph Pn+1 np is graceful. Theorem.: The sun extension graph G = (P Pn+1) e is graceful, where e is an edge having end vertices of each degree.

4 1360 A. Solairaju, S. Subbulakshmi and R. Kokila Proof: The given G has the vertex set {v 1, v,, v n+ } and the edge set {(v i v i+1 ); i = 1 to n 1; i n} {v j v n+j ; i = 1 to n} {v n v n+1 } {v n v n+ }. One of the arbitrary labelings of the graph G for vertices is as follows (figure ): Define f: V(G) {0,1,, q} as follows: Case 1: n is odd f(v n+1 i ) = q ( i 1 ) ; i = 1,3,5,, n. f(v i ) = q ( i ) ; i = n + 1, n + 3,.., (n ). f(v n+1 i ) = i ; i =,4,, (n 1). f(v i ) = ( i+1 ) ; i = n +, n + 4,, (n 1). f(v n ) = 0; f(v n+1 ) = f(v n 1 ) + 1; f(v n+ ) = f((v n )) 1. Case : n is even f(v n+1 i ) = q ( i 1 ) where i varies from 1 to n and i is odd. f(v n+1 i ) = i ; i varies from 1 to n and i is even. f(v i ) = q ( i ) where i = (n + 1), (n + 3),, (n ) (i is odd). f(v i ) = ( i+1 ) ; i = (n + 1),, (n ) (i is even). f(v n ) = 0; f(v n+1 ) = f(v n 1 ) + 1; f(v n+ ) = f((v n ) 1. Its induced map f + : E(G) {1,,, q} is defined by f + (uv) = f(u) f(v) for all edge uv in E(G). Here f + is an injective map. Both f and f + satisfy the condition for graceful labeling. Hence G (P Pn+1) e is graceful.

5 Graceful Labelings for Sun Graph, Sun Extension Graph, Double Fan, Theorem.3: The double fan F(nC3) is graceful. Proof: The given graph has vertex set = {v 0, v 1, v,. v 3n 1, V 3n } and edge set is { v0vi: i = 1 to 3n} { vivn+i : I = 1 to n} { vn+ivn+i : I = to (n-1)} { v1v3n}. One of the arbitrary labelings of the graph G for vertices are as follows (figure 3): Define f: V(G) {0,1,, q 1} by f(v i ) = 5n i + 1; i = 1,,, n. f(v i ) = (3n + 1) + (i n 1) = n + i; i = n + 1, n +,, n. f(v i ) = (3n 1) + (i n 1) = n + i ; i = n + 1,,3n. Its induced map f + : E(G) {1,,..., q} is defined by f + (uv) = f(u) f(v) for all edge uv in E(G). Here f + is an injective map. Both f and f + satisfy the condition for graceful labeling. The double fan F(nC3) is graceful. Definition.4: The connected graph Jn, 4 is obtained from a path Pn in which each vertex is identified with any pendent vertex of the path P4. Definition.5: The connected graph Jn, 4 S3 is obtained from a path Pn in which each vertex is identified with any one of pendent vertices of the path P4, and the remaining pendent of P4 is merged with the root (center) of the star S3. Theorem.6: The connected graph (leg graph with star) Jn, 4 S3 is graceful. Proof: Its vertex set is {v1, v,, v7n}, and edge set is {vivi+1: i = 1 to n-1} {vivn+i, vn+ivn+i, vn+iv3n+i, : i = 1 to n} { v3n+iv4n+i, v3n+iv5n+i, v3n+iv6n+i: i = 1 to n} ( figure 4).

6 136 A. Solairaju, S. Subbulakshmi and R. Kokila Define f: V(G) {0,1,.,, q} by f(v i ) = ( i 1 ) 7 i is odd, i = 1 to n = (p 1) ( i 1 ) 7 (i n) is odd, i = (n + 1),.n = 1 + ( i 1 ) 7 (i n) is odd, i = (n + 1),,3n = (p ) ( i 1 ) 7 ( i 3n) is odd, i = (3n + 1),,4n. = + ( i 1 ) 7 (i 4n) is odd, i = (4n + 1),, 5n. = 3 + ( i 1 ) 7 (i 5n) is odd, i = (5n + 1),,6n. = 4 + ( i 1 ) 7 (i 6n) is odd, i = (6n + 1),,7n. f(v i ) = (p 7) ( i ) 7 i is even, i = 1,, n. = 6 + ( i ) 7 ( i n)is even, i = (n + 1),, n. = (p 6) ( i ) 7 (i n)is even, i = (n + 1),, 3n. = 5 + ( i ) 7 (i 3n) is even, i = (3n + 1),, 4n. = (p 3) ( i ) 7 ( i 4n)is even, i = (4n + 1),, 5n. = (p 4) ( i ) 7 (i 5n)is even, i = (5n + 1),, 6n. = (p 5) ( i ) 7 (i 6n)is even, i = (6n + 1),, 7n..

7 Graceful Labelings for Sun Graph, Sun Extension Graph, Double Fan, The induced map f + : E(G) {1,,..., q} is defined by f + (uv) = f(u) f(v) for all edge uv in E(G). Here f + is an injective map. Both f and f + satisfy the condition for graceful labeling. The connected graph (Leg graph) Jn, 4 S3 is graceful. REFERENCES [1] Abhyankar V. J, Direct methods of gracefully labeling graphs, Ph.D. thesis, University of Mumbai, 00. [] Barrientos C, Equitable labelings of corona graphs, J. Combin. Math. Combin. Comput., Volume 41, (00), [3] Bermond, E., Brouwer A. E, and Germa A, Systemes de triplets et differences associ`ees, Problems Combinatories et Th`eorie des Graphs, Colloq. Intern. du Centre National de la Rech. Scient., 60, Editions du Centre Nationale de la Recherche Scientifique, Paris (1978) [4] Bhat Nayak.V.N.,& Selvam,A, Gracefulness of n-cone C m + K m c, Ars Combin., Volume 66 (003), [5] Bodendiek R, Schumacher H, and Wegner H, Uber eine spezielle Klasse grozi oser Eulerscher Graphen, Mitt. Math. Gesellsch. Hamburg, Volume 10, (1975), [6] Gallian, J.A., A survey: recent results, conjectures and open problems on labeling graphs, Journal of Graph Theory, Volume 13, (1989), [7] Graham, R.L and Sloane N.J.A, On additive bases and harmonious graphs, SIAM Journal of Algebra Discrete Math, Volume 1, (1980), [8] Guo W. F, Gracefulness of the graph B (m, n), J. Inner Mongolia Normal Univ., (1994), 4 9. [9] Huang.J and Skiena.S, Gracefully labeling prisms, Ars Combin., 38, (1994), 5-4. [10] Koh, K. M., Rogers, D. G., Lee, P. Y., and Toh, C.W., On graceful graphs V: unions of graphs with one vertex in common, Nanta Math., Volume 1, (1979), [11] Koh, K. M., Rogers, D. G., Teo, H. K., and Yap, K.Y., Graceful graphs: some further results and problems, Congr. Numer., Volume 9, (1980), [1] Lee., S.M, and Seah., E, On edge-gracefulness of kth power cycles, Congr. Numer., Volume 71, (1990), [13] Liu,Y., Gracefulness of a star with top sides, J.Sichuan Normal Univ.,18(1995) [14] Ma, K.J., & Feng, C.J., On gracefulness of gear graphs, Math. Practice Theory, (1984) 7-73.

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