Journal of Computer and Mathematical Sciences, Vol.6(2),127-133, February 2015 ISSN 0976-5727 (Print) (An International Research Journal), www.compmath-journal.org ISSN 2319-8133 (Online) Ǧ -Graceful Labeling of Some Graphs Shigehalli V. S. and Chidanand A. Masarguppi Professor, Department of Mathematics, Ranichannmma University, Vidyasangama, Belagavi, INDIA. (Received on: January 24, 2015) ABSTRACT Let G be a graph .The Ǧ -graceful labeling of a graph G (p , q) with p vertices and q edges is a injective a function f : V (G) Ŵ {0, 1, 2, --- - - -n-1} such that the induced function f* : E (G) Ŵ N is given by f* (u ,v) = 2 { f (u) +f (v) }, the resulting edge labels are distinct .In this paper we prove result on Ǧ -graceful labeling of cyclic graph, Paths, ladder graphs, flower graph, K2,n friendship graph. Keywords: Cyclic graphs, ladder graphs, path, flower graph friendship graph, K 2,n . 1. INTRODUCTION Graph labeling is an active area of research in graph theory. Graph labeling where the vertices are assigned some value subject to certain condition. Labeling of vertices and edges play a vital role in graph theory .To begin with simple, finite graph G = (V(G),E(G) ) with p vertices and q edges. The definitions and other information which are used for the present investigations are given. 2. DEFINITIONS Definition 2.1: Ǧ-gracefull graph: A function f is called a Ǧ-graceful labeling of graph G if f : V ( G ) Ŵ {0, 1, 2, 3, ---------------------n-1} is injective and the induced function f * : E ( G ) N is defined as f* ( e = uv ) = 2 { f (u) + f (v) },then edge labels are distinct. Definition 2.2: flower graph: Let G be a graph with order ‘n’ and size n-1, such that exactly one node is adjacent to every other n-1 nodes. The resulting graph is flower graph with n -1 petal. February, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 128 Shigehalli V. S., et al., J. Comp. & Math. Sci. Vol.6 (2), 127-133 (2015) Definition 2.3: Ladder graph : The ladder L n ( n ≥ 2 ) is the product graph P 2 Ɛ Pn which contains 2n vertices and 3n-2 edges. Definition 2.4: Friendship graph: A friendship graph F n is a graph which consists of n triangles with a common vertices. 3. RESULTS Theorem 3.1 : C n is a Ǧ Ǝgraceful graph if ‘n’ is odd. Proof: Let C n be a cycle with vertex set {v 1 ,v 2 ,v 3 , v4---vn } and edge set {e 1,e 2,e 3----en }. We define the vertex labeling f : V( G ) { 0, 1, 2 ,-----n-1 } as f ( v 1 ) = 0 f ( v 2 ) = 1. : : : f ( v n ) = n-1, label the vertex in both direction. The edge labeling function f* is defined as follows f* : E ( G ) N is defined by f* ( u v ) = 2 { f (u) + f (v) } f* (v 1vn ) = 2 { f (v 1) + f (v n) } : : = {2 ,6, 10, -------- (4n-6) } The edge labels are distinct. Thus f is Ǧ - graceful labeling of G = C n Hence C n is a Ǧ - graceful graph when ‘n’ is odd. Illustration: Ǧ - graceful labeling of the graph C 5 is shown in fig 1 0 8 2 4 1 14 6 3 10 2 Fig1. Ǧ - Graceful graph of C 5 February, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org Shigehalli V. S., et al., J. Comp. & Math. Sci. Vol.6 (2), 127-133 (2015) 129 Theorem 3.2 : Ladders ( L n ) are Ǧ - graceful graph if ‘n’ is even. Proof : Let G = L n be the ladder on 2n vertices. Let v 1,v 2,v 3 -------- vn be vertices of one path and u 1,u 2, u3 ---------un be the vertices of another path. Label the vertices from left side of one path by x, x+1, x+2,-------x+(n-1) (x=0) and the other vertices of the path again from left side of path by x + n , x + (n+1), -------- x+(2n-1). v 1 v 2 v3 v 4 v 5 P1 u 1 u 2 u3 u 4 u 5 P2 The edge labeling function f* is defined as follows f* : E ( G ) N is defined by f* ( u v ) = 2 { f (u) + f (v) }such that the edge labels are distinct. In view of the above labeling pattern the ladders are Ǧ - graceful labeling. Hence L n is a Ǧ - graceful graph if n is even . Illustration: Ǧ - Graceful labeling of the graph L 4 is shown in fig 2 0 2 1 6 2 10 3 8 12 16 20 4 18 5 22 6 26 7 Fig2. Ǧ - Graceful graph of L 4 Theorem 3.3: Paths ( P n ) are Ǧ -graceful graphs. Proof: Let P n be a path with vertex set v 1,v 2, v3, v4, v5, v6, v7, --- vn and e 1, e 2, e 3 , --- en-1 be the edges of the path. February, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 130 Shigehalli V. S., et al., J. Comp. & Math. Sci. Vol.6 (2), 127-133 (2015) We define the vertex labeling f : V ( G ) Ŵ { 0, 1, 2, -------n-1 } f ( v 1 ) = 0 f ( v 2 ) =1. : : f ( v n ) = n-1 , vertex labeling can be done in both direction. The edge labeling function f* is defined as follows Ŵ f* : E ( G ) N is defined by f* ( u v ) = 2 { f ( u ) + f (v) } f ( v v n ) = 2 { f (v) + f (v n) } : : = { 2 , 6, 10 ,----- (4n-6) } such that the edge labels are distinct. In view of above labeling pattern the paths are Ǧ - graceful labeli ng. Hence P n is a Ǧ - graceful graph. Illustration: Ǧ - Graceful labeling of the graph P 6 is shown in fig 3 2 6 10 14 18 0 1 2 3 4 5 Fig3. Ǧ - Graceful graph of P 6 Theorem 3.3 : Flower graphs are Ǧ - graceful graphs. Proof: Let G be a flower graph. Then G has n vertices and (n-1) edges. Therefore V = { v 1 , v 2 ,---- vn } and E = { e 1, e 2,- --en-1 } Ŵ We define the vertex labeling f : V ( G ) { 0, 1, 2, ---n-1 } f ( v 1 ) = 0 f ( v 2 ) = 1. : : f ( v n ) = n -1 such that labeling of the vertices may be clockwise or anticlockwise. The edge labeling function f* is defined as follows f* : E ( G ) Ŵ N is defined by f* ( u v ) = 2 { f (u) + f (v) } f ( v 1vn) = 2 { f (v 1) + f (v n) } February, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org Shigehalli V. S., et al., J. Comp. & Math. Sci. Vol.6 (2), 127-133 (2015) 131 : : = {2, 4, 6 ,--------2n} then the edge labels are distinct and is in increasing order. Hence all flower graphs are Ǧ - graceful graph . Illustration: Flower graph with 6 petals in fig 4 . 4 5 3 2 6 8 6 1 12 10 4 2 0 Fig4. Ǧ - Graceful graph of flower graph. Theorem 3.5 : A friendship Graph ( F 2) is a Ǧ - graceful graph. Proof: Let G = F n be the friendship Graph. Let v 1, v 2, v 3,---- vn be the vertices of F n and edges are e 1, e 2, e 3, e 4,--- em. We define the vertex labeling f : V(G) {0,1,2,---n-1} f (v 1) = 0 f (v 2)=1. : : f ( v n ) = n-1. The edge labeling function f* is defined as follows f*: E(G) N is defined by f* ( uv ) = 2 { f (u) + f (v) } f * (v 1v2) = 2 { f ( v 1 ) + f ( v 2 ) } = 2 ; : : = {2, 4, 6, 10, 14, 12 } The edge labels are distinct In view of the above labeling pattern the F2 is a Ǧ - graceful labeling. Hence F 2 is a Ǧ - graceful graph. February, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 132 Shigehalli V. S., et al., J. Comp. & Math. Sci. Vol.6 (2), 127-133 (2015) Illustration: Ǧ - Graceful labeling of the graph F 2 is shown in fig 5 0 2 1 4 6 2 12 10 4 14 3 Fig 5. Ǧ - Graceful graph of F 2 graph Theorem 3.6 : Complete Bipartite graph (K 2,n ) is a Ǧ - graceful graph. Proof: Let G = K 2,n be a complete Bipartite graph. Let the vertex set be v 1, v 2 , v 3- ---- vn , vn + 1 , vn + 2 and K 2,n has 2 Ɛ n number of edges. The vertex set is partitioned into two sets V 1 & V 2 where V 1 = { v 1 ,v 2 } and V2 = { v 3----- vn , vn + 1 , vn + 2 }. Define f: V(G) {0,1,2,---n-1} such that f(v1) = 0 f(v 2)=1. : : f ( v n ) = n-1 Label the vertex in both direction and vertex is fixed in top of the graph continuing in this fashion until all the vertices are labeled. The edge labeling function f* is defined as follows f* : E(G) N is defined by f* ( uv ) = 2 { f (u) + f (v) } : : = { 2, 4, 6, .----2n } The edge labels are distinct. Hence K 2, n is a Ǧ - graceful graph. February, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org Shigehalli V. S., et al., J. Comp. & Math.