ISSN 2319-8133 (Online)
(An International Research Journal), www.compmath-journal.org
-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 -graceful labeling of a graph G (p , q)with p vertices and q edges is a injective a function f : V (G) {0, 1, 2, --- - - -n-1} such that the induced function f* : E (G) 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 -graceful labeling of cyclic graph, Paths, ladder graphs, flower graph, K2,n friendship graph.
Keywords: Cyclic graphs, ladder graphs, path, flower graph friendship graph, K2,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: -gracefull graph: A function f is called a -graceful labeling of graph G if f : V ( G ) {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 toevery other n-1 nodes. The resulting graph is flower graph with n -1 petal.
Definition 2.3: Ladder graph: The ladder Ln ( n ≥ 2 ) is the product graph P2 Pn which contains 2n vertices and 3n-2 edges.
Definition 2.4: Friendship graph: A friendship graph Fn is a graph which consists of n triangles with a common vertices.
3. RESULTS
Theorem 3.1: Cn is a graceful graph if ‘n’ is odd.
Proof: Let Cn be a cycle with vertex set {v1 ,v2 ,v3 ,v4---vn } and edge set {e1,e2,e3----en }.
We define the vertex labeling f : V( G ) { 0, 1, 2 ,---n-1 } as f ( v1 ) = 0
f ( v2 ) = 1.
: : :
f ( vn ) = 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* (v1vn ) = 2 { f (v1) + f (vn) }
: :
= {2 ,6, 10, --- (4n-6) }
The edge labels are distinct.
Thus f is - graceful labeling of G = Cn Hence Cn is a - graceful graph when ‘n’ is odd.
Illustration:
- graceful labeling of the graph C5 is shown in fig 1 0
8 2
4 1
14 6
3 10 2
Fig1. - Graceful graph of C5
129
Theorem 3.2 : Ladders ( Ln ) are - graceful graph if ‘n’ is even.
Proof : Let G = Ln be the ladder on 2n vertices.
Let v1,v2,v3 --- vn be vertices of one path and u1,u2,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 - graceful labeling.
Hence Ln is a - graceful graph if n is even . Illustration:
- Graceful labeling of the graph L4 is shown in fig 2
0 2 1 6 2 10 3
8 12 16 20
4 18 5 22 6 26 7
Fig2. - Graceful graph of L4
Theorem 3.3: Paths ( Pn ) are -graceful graphs.
Proof: Let Pn be a path with vertex set v1,v2,v3,v4,v5,v6, v7, --- vn and e1, e2, e3 ,--- en-1 be the
edges of the path.
We define the vertex labeling f : V ( G ) { 0, 1, 2, ---n-1 } f ( v1 ) = 0
f ( v2 ) =1.
: : f ( vn ) = n-1 , vertex labeling can be done 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 vn ) = 2 { f (v) + f (vn) }
: :
= { 2 , 6, 10 ,--- (4n-6) } such that the edge labels are distinct.
In view of above labeling pattern the paths are - graceful labeling.
Hence Pn is a - graceful graph.
Illustration:
- Graceful labeling of the graph P6 is shown in fig 3
2 6 10 14 18
0 1 2 3 4 5 Fig3. - Graceful graph of P6
Theorem 3.3 : Flower graphs are - graceful graphs.
Proof: Let G be a flower graph. Then G has n vertices and (n-1) edges.
Therefore V = { v1 , v2 ,---- vn } and E = { e1, e2,- --en-1 } We define the vertex labeling f : V ( G ) { 0, 1, 2, ---n-1 } f ( v1 ) = 0
f ( v2 ) = 1.
: :
f ( vn ) = 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 ) N is defined by f* ( u v ) = 2 { f (u) + f (v) }
f ( v1vn) = 2 { f (v1) + f (vn) }
131 : :
= {2, 4, 6 ,---2n} then the edge labels are distinct and is in increasing order.
Hence all flower graphs are - 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. - Graceful graph of flower graph.
Theorem 3.5: A friendship Graph ( F2) is a - graceful graph.
Proof: Let G = Fn be the friendship Graph.
Let v1, v2, v3,---- vn be the vertices of Fn and edges are e1, e2, e3, e4,--- em. We define the vertex labeling f : V(G) {0,1,2,---n-1}
f (v1) = 0 f (v2)=1.
: :
f ( vn ) = 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 * (v1v2) = 2 { f ( v1 ) + f ( v2 ) } = 2 ;
: :
= {2, 4, 6, 10, 14, 12 } The edge labels are distinct
In view of the above labeling pattern the F2 is a - graceful labeling.
Hence F2 is a - graceful graph.
Illustration:
- Graceful labeling of the graph F2 is shown in fig 5 0 2 1
4 6
2 12 10
4 14 3
Fig 5. - Graceful graph of F2 graph
Theorem 3.6 : Complete Bipartite graph (K2,n) is a - graceful graph.
Proof: Let G = K2,n be a complete Bipartite graph.
Let the vertex set be v1, v2 , v3- ---- vn , vn + 1 ,vn + 2 and K2,n has 2 n number of edges.
The vertex set is partitioned into two sets V1 & V2 where V1 = { v1 ,v2 } and V2 = { v3--- vn , vn + 1 ,vn + 2 }.
Define f: V(G) {0,1,2,---n-1} such that f(v1) = 0
f(v2)=1.
: :
f ( vn ) = 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 K2, n is a - graceful graph.
133 Illustration:
- Graceful labeling of the graph K2,3 is shown in fig 6
4 0
6
14 12 10 4 2
3 2 1 Fig5. - Graceful graph of K 2,3 graph
CONCLUSION
In this paper we have shown that every ladder graphs are - Graceful graph, every path graphs are - Graceful graph, friendship graph is a - Graceful graph, every flower graph is a - Graceful graph. Are investigated it can also verified for some graphs.
REFERENCES
1. A Gillian. A dynamic Survey of graph labeling, the Electronics Journal of Combinatories Dec 20, (2013).
2. Frank Harray. Graph Theory, Narosa Publishing House .
3. Gray Chatrand, Ping Zhang, Introduction to graph theory, McGraw –Hill international Edition.
4. Murugesan N, Uma.R ,Graceful labeling of some graphs and their sub graphs Asian Journal of Current Engineering and Maths 6 Nov-Dec. (2012).
