Prime Cordial Labeling For Some Cycle Related Graphs

Transcription

1 Int. J. Open Probles Copt. Math., Vol. 3, No. 5, Deceber 010 ISSN ; Copyright c ICSRS Publication, Prie Cordial Labeling For Soe Cycle Related Graphs S K Vaidya 1 and P L Vihol 1 Saurashtra University, Rajkot , GUJARAT (INDIA) sairkvaidya@yahoo.co.in Governent Engineering College, Rajkot , GUJARAT (INDIA) viholprakash@yahoo.co Abstract We present here prie cordial labeling for the graphs obtained by soe graph operations on cycle related graphs. Keywords: prie cordial labeling, prie cordial graph, duplication, path union, friendship graph. 1 Introduction If the vertices of the graph are assigned values subject to certain conditions is known as graph labeling. A dynaic survey on graph labeling is regularly updated by Gallian [] and it is published by Electronic Journal of Cobinatorics. Vast aount of literature is available on different types of graph labeling and ore than 1000 reserch papers have been published so far in past three decades. For any labeling probles following three characteristics are really noteworthy A set of nubers fro which vertex labels are chosen; A rule that assigns a value to each edge; A condition that these values ust satisfy. The present work is aied to discuss one such a labeling naely prie cordial labeling. We begin with siple,finite,connected and undirected graph G = (V (G), E(G))

2 4 S.K. Vaidya et al. with p vertices and q edges. For all other terinology and notations in graph theory we follow West [6]. We will give brief suary of definitions which are useful for the present investigations. Definition 1.1 Let G = (V (G), E(G)) be a graph. A apping f : V (G) {0, 1} is called binary vertex labeling of G and f(v) is called the label of the vertex v of G under f. For an edge e = uv, the induced edge labeling f : E(G) {0, 1} is given by f (e) = f(u) f(v). Let v f (0), v f (1) be the nuber of vertices of G having labels 0 and 1 respectively under f and let e f (0), e f (1) be the nuber of edges having labels 0 and 1 respectively under f. Definition 1. A binary vertex labeling of a graph G is called a cordial labeling if v f (0) v f (1) 1 and e f (0) e f (1) 1. A graph G is cordial if it adits cordial labeling. This concept was introduced by Cahit [1] as a weaker version of graceful and haronious graphs. After this any researchers have investigated graph failies or graphs which adit cordial labeling. Soe labeling schees are also introduced with inor variations in cordial thee. Soe of the are product cordial labeling, total product cordial labeling and prie cordial labeling. The present work is focused on prie cordial labeling which is defined as follows. Definition 1.3 A prie cordial labeling of a graph G with vertex set V (G) is a bijection f : V (G) {1,, 3,..., p} defined by f(e = uv) = 1; if gcd(f(u), f(v)) = 1 = 0; otherwise and e f (0) e f (1) 1. A graph which adits prie cordial labeling is called a prie cordial graph. The concept of prie cordial labeling was introduced by Sundara et al.[4] and in the sae paper they investigate several results on prie cordial labeling. Vaidya and Vihol [5] have also discussed prie cordial labeling in the context of graph operations. In the present work we will investigate soe new prie cordial graphs. Definition 1.4 Duplication of a vertex v k by a new edge e = v kv k graph G produces a new graph G such that N(v k) N(v k) = v k. in a Definition 1.5 Duplication of an edge e = uv by a new vertex w in a graph G produces a new graph G such that N(w) = {u, v}.

3 Prie Cordial Labeling For Soe 5 Definition 1.6 (Shee and Ho[3]) Let graphs G 1, G,..., G n, n be all copies of a fixed graph G. Adding an edge between G i to G i+1 for i = 1,,..., n 1 is called the path union of G. Definition 1.7 A Friendship graph F n is a one point union of n copies of cycle C 3. Main Results Theore.1 The graph obtained by duplicating each edge by a vertex in cycle C n adits prie cordial labeling except for n = 4. Proof: If C n be the graph obtained by duplicating an edge by a vertex in a cycle C n then let v 1, v,..., v n be the vertices of cycle C n and v 1, v,..., v n be the added vertices to obtain C n corresponding to the vertices v 1, v,..., v n in C n. Define f : V (C n) {1,, 3..., p}, we consider following two cases. Case 1: n is odd Sub Case 1: n = 3, 5 The prie cordial labeling of C n for n = 3, 5 is as shown in Figure 1. Fig 1 Prie cordial labeling of C 3 and C 5 Sub Case : n 7 f(v 1 ) =, f(v ) = 4, f(v +i ) = 6 + i; 1 i n f(v n+1 ) = 6, f(v n+1 +1) = 1, f(v n ++i ) = 4i + 3; 1 i n 1 f(v i) = f(v n ) + i; 1 i n f(v n +1) = 3, f(v n +1+i) = 4i + 1; 1 i n

4 6 S.K. Vaidya et al. In the view of the labeling pattern defined above we have e f (0) + 1 = e f (1) = 3 n + Case : n is even Sub Case 1: n = 4 For the graph C 4 the possible pairs of labels of adjacent vertices are (1, ), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (, 3), (, 4), (, 5), (, 6), (, 7), (, 8), (3, 4), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8), (5, 6), (5, 7), (5, 8), (6, 7), (6, 8), (7, 8). Then obviously e f (0) = 5, e f (1) = 7. That is, e f (1) e f (0) = and in all other possible arrangeent of vertex labels e f (0) e f (1). Thus C 4 is not a prie cordial graph. Sub Case : n = 6, 8, 10 The prie cordial labeling of C 6, C 8 and C 10 is as shown in Figure Fig Prie cordial labeling of C 6, C 8 and C Sub Case 3: n 1 f(v 1 ) =, f(v ) = 4, f(v 3 ) = 8, f(v 4 ) = 10, f(v 5 ) = 14, f(v 5+i ) = 14 + i; 1 i n 6 f(v n ) = 6, f(v n+1 +1) = 3, f(v n +1+i ) = 4i + 1; 1 i n 1 f(v 1) = n f(v 1+i) = f(v n 1 ) + i; 1 i n ) = 1 f(v n f(v n +1) = 1 f(v n +1+i) = 4i + 3; 1 i n 1 In the view of the labeling above defined we have e f (0) = e f (1) = 3n Thus in the above two cases we have e f (0) e f (1) 1 Hence the graph obtained by duplicating each edge by a vertex in a cycle C n adits prie cordial labeling except for n = 4.

5 Prie Cordial Labeling For Soe 7 Exaple. Consider the graph C 1. The labeling is as shown in Figure 3. Fig 3 Prie cordial labeling of C 1 Theore.3 The graph obtained by duplicating a vertex by an edge in cycle C n is prie cordial graph. Proof: If C n be the graph obtained by duplicating a vertex by an edge in cycle C n then let v 1, v,..., v n be the vertices of cycle C n and v 1, v,..., v n be the added vertices to obtain C n corresponding to the vertices v 1, v,..., v n in C n. To define f : V (C n) {1,, 3..., 3p}, we consider following two cases. Case 1: n is odd Sub Case 1: n = 3, 5 The prie cordial labeling of C n for n = 3, 5 is shown in Figure 4. Fig 4 Prie cordial labeling of C 3 and C 5

6 8 S.K. Vaidya et al. Sub Case : n 7 f(v 1 ) =, f(v ) = 4, f(v +i ) = 6 + i; 1 i n f(v n+1 ) = 3, f(v n+1 +1) = 1, f(v n ++i ) = 6i + 5; 1 i n 1 f(v i) = f(v n ) + i; 1 i n f(v n +1) = 6, f(v n +) = 9 f(v n +3) = 5, f(v n +4) = 7 f(v n +4+i 1) = 6i + 7; 1 i n 1 f(v n +4+i) = 6i + 9; 1 i n 1 Case : n is even Sub Case 1: n = 4, 6 The prie cordial labeling of C n for n = 4, 6 is shown in Figure 5. Fig 5 Prie cordial labeling of C 4 and C 6 Sub Case : n 8 f(v 1 ) =, f(v ) = 4, f(v +i ) = 6 + i; 1 i n 3 f(v n ) = 6, f(v n +1 ) = 3, f(v n +1+i ) = 6i + 1; 1 i n 1 f(v i) = f(v n 1 ) + i; 1 i n f(v n+1) = 1, f(v n+) = 5 f(v n+1+i) = 6i + 3; 1 i n 1 f(v n++i) = 6i + 5; 1 i n 1 Thus in both the cases defined above we have e f (0) = e f (1) = n Hence C n adits prie cordial labeling.

7 Prie Cordial Labeling For Soe 9 Exaple.4 Consider the graph C 7. The labeling is as shown in Figure 6. Fig 6 Prie cordial labeling of C 7 Theore.5 The path union of copies of cycle C n is a prie cordial graph. Proof: Let G be the path union of copies of cycle C n and v 1, v, v 3, v 4...,v n be the vertices of G. To define f : V (G ) {1,, 3..., n} we consider following four cases. Case 1: n even, even f(v i ) = i; f(v n +1 ) = 1, 1 i n f(v n +1+i ) = 4i 1; 1 i n f(v n + n + ) = f(v n + n +1 ), f(v n + n ++i ) = f(v n f(v n +jn+i ) = f(v n Case : n odd, even f(v i ) = i; f(v n +1 ) = 1, f(v n + n + ) 4i; 1 i n +(j 1)n+i ) + n; 1 j 1, 1 i n 1 i n +1+i ) = 4i 1; 1 i n f(v n + n + ) = f(v n + n +1 ) +, f(v n + n ++i ) = f(v n + n + ) 4i; 1 i n 1 f(v n +jn+i ) = f(v n +(j 1)n+i ) + n; 1 j 1, 1 i n using above pattern we have e f (0) + 1 = e f (1) = (n+1) Case 3: n even, odd f(v 1 ) = 4, f(v ) = 8, f(v +i ) = 8 + i; 1 i n +1 ) =

8 30 S.K. Vaidya et al. +1+i ) = ) + i; 1 i n + n ) = 6, + n +1 ) = 3, + n + ) = 1 + n ++i ) = i i n +n+1 ) = +n ) + or +n+1 ) = +n+ ) = +n+1 ) +, +n++i ) = +n+ n + ) = +n+ n ++i ) = +n+1 ) + 4i, 1 i n 1 +n+ n +1 ) + +n+ n + ) 4i; 1 i n +n ) + 4 for n = 4 +(j+1)n+i ) = f(v n +(j)n+i ) + n; 1 j 1, 1 i n using above pattern we have e f (0) = e f (1) = (n + 1) + n Case 4: n odd, odd Sub Case 1: n = 3 f(v 1 ) =, f(v ) = 4, f(v +i ) = 6 + i; 1 i n +1 ) = 6, + ) = 3, +3 ) = 5, +4 ) = 1, +4+i ) = i + 3; 1 i n 1 using above pattern we have e f (0) + 1 = e f (1) = (n + 1) + Sub Case : n 5 f(v 1 ) = 4, f(v ) = 8, f(v +i ) = 8 + i; 1 i n +1 ) =, +1+i ) = + n ) = 6, ) + i; 1 i n + n +1 ) = 3, + n + ) = 1, + n ++i ) = i + 3, 1 i n 1 +n+1 ) = +n ) +, f(v n +n+1+i ) = +n+1 ) + i, 1 i n 1 f(v n +(j+1)n+i ) = f(v n +(j)n+i ) + n; 1 j 1, 1 i n using above pattern we have e f (0) + 1 = e f (1) = (n + 1) + n + 1 Thus in all the above cases we have e f (0) e f (1) 1. Hence G adits prie cordial labeling.

9 Prie Cordial Labeling For Soe 31 Exaple.6 Consider the path union of three copies of C 7. The labeling is as shown in Figure 7. Fig 7 Prie cordial labeling of C 7 Theore.7 The friendship graph F n is a prie cordial graph for n 3. Proof: Let v 1 be the vertex coon to all the cycles. Without loss of generality we start the label assignent fro v 1. To define f : V (F n ) {1,, 3..., n + 1}, we consider following two cases. Case 1: n even let p be the highest prie such that 3p n + 1, f(v 1 ) = p, now label the reaining vertices fro 1 to n + 1 except p. In the view of the labeling pattern defined above we have e f (0) = e f (1) = 3n Case : n odd let p be the highest prie such that p n + 1, f(v 1 ) = p, now label the reaining vertices fro 1 to n + 1 except p. In the view of the labeling above defined we have e f (0) + 1 = e f (1) = 3 n + Thus in above two cases e f (0) e f (1) 1 Hence friendship graph adits prie cordial labeling. Exaple.8 Consider the friendship graph F 8. The labeling is as shown in figure 8. Fig 8 Prie cordial labeling of F 8

10 3 S.K. Vaidya et al. 3 Concluding Rearks Labeling of discrete structure is a potential area of research due to its diversified applications and it is very interesting to investigate whether any graph or graph faily adit a particular labeling or not? Here we contribute four results in the context of prie cordial labeling. Shee and Ho[3] have proved that the path union of cycles adits cordial labeling while we show that the path union of cycles adits prie cordial labeling. 4 Open Proble Analogous results can be investigated for various graph failies. Siilar results can be obtained in the context of different graph labeling techniques 5 Acknowledgeent The authors are highly thankful to anonyous referee for valuable coents and kind suggestions. References [1] I Cahit, Cordial Graphs: A weaker version of graceful and haronious graphs, Ars Cobinatoria, 3(1987), [] J A Gallian, A dynaic survey of graph labeling, The Electronics Journal of Cobinatorics, 17(010) #DS6. [3] S C Shee, Y S Ho, The Cordiality of the path-union of n copies of a graph, Discrete Math., 151(1996), 1-9. [4] M Sundara, R Ponraj and S Soasundara, Prie cordial labeling of graph, J.Indian Acad.Math., 7(), [5] S K Vaidya and P L Vihol, Prie cordial labeling for soe graphs, Modern Applied Science, 4(8)(010), [6] D B West, Introduction To Graph Theory, Prentice-Hall, India, 001.