cordial and 3-equitable labeling for some wheel …cordial and 3-equitable labeling for some wheel...

Cordial and 3-equitable Labeling for Some WheelRelated Graphs

S K Vaidya , N A Dani , K K Kanani , P L Vihol

Abstract—We present here cordial and 3-equitable labelingfor the graphs obtained by joining apex vertices of two wheelsto a new vertex. We extend these results for k copies of wheels.

Index Terms—Cordial graph, Cordial labeling, 3-equitablegraph, 3-equitable labeling

AMS Subject classification number(2000): 05C78.

I. INTRODUCTION

WE begin with simple, finite and undirected graphG = (V,E). In the present work Wn = Cn + K1

(n ≥ 3) denotes the wheel and in Wn vertices correspond toCn are called rim vertices and vertex which corresponds toK1 is called an apex vertex. For all other terminology andnotations we follow Harary [7]. We will give brief summaryof definitions which are useful for the present investigations.

Definition 1.1 Consider two wheels W(1)n and W

(2)n

then G =< W(1)n : W

(2)n > is the graph obtained by joining

apex vertices of wheels to a new vertex x.Note that G has 2n+ 3 vertices and 4n+ 2 edges.

Definition 1.2 Consider k copies of wheelsnamely W

(1)n ,W

(2)n ,W

(3)n , . . .W

(k)n . Then the

G =< W(1)n : W

(2)n : W

(3)n : . . . : W

(k)n > is the

graph obtained by joining apex vertices of each W(p−1)n and

W(p)n to a new vertex xp−1 where 2 ≤ p ≤ k.

Note that G has k(n+2)−1 vertices and 2k(n+1)−2 edges.

Definition 1.3 If the vertices of the graph are assignedvalues subject to certain conditions then it is known asgraph labeling.

According to Hegde [8] most interesting graph labelingproblems have following three important characteristics.

1) a set of numbers from which the labels are chosen;2) a rule that assigns a value to each edge;3) a condition that these values must satisfy.The recent survey on graph labeling can be found in

Gallian [6]. Vast amount of literature is available on differenttypes of graph labeling. According to Beineke and Hegde[2] graph labeling serves as a frontier between numbertheory and structure of graphs.

Labeled graph have variety of applications in codingtheory, particularly for missile guidance codes, design of

S K Vaidya is with the Saurashtra University, Rajkot, 360005 IN-DIA(phone:+919825292539; e-mail: [email protected])

N A Dani is with Government Polytechnic, Junagadh, 362001 INDIA(e-mail: nilesh a [email protected])

K K Kanani is with L E College, Morbi, 363642 INDIA (e-mail:[email protected])

P L Vihol is with Government Polytechnic, Rajkot, 360002 INDIA (e-mail: [email protected])

good radar type codes and convolution codes with optimalautocorrelation properties. Labeled graph plays vital role inthe study of X-Ray crystallography, communication networkand to determine optimal circuit layouts. A detailed studyon variety of applications of graph labeling is carried out inBloom and Golomb [3].

Definition 1.4 Let G = (V,E) be a graph. A mappingf : V (G) →{0,1} is called binary vertex labeling of G andf(v) is called the label of the vertex v of G under f .

For an edge e = uv, the induced edge labelingf∗ : E(G) → {0, 1} is given by f∗(e)=|f(u) − f(v)|. Letvf (0), vf (1) be the number of vertices of G having labels0 and 1 respectively under f and let ef (0),ef (1) be thenumber of edges having labels 0 and 1 respectively under f∗.

Definition 1.5 A binary vertex labeling of a graph Gis called a cordial labeling if |vf (0) − vf (1)| ≤ 1 and|ef (0) − ef (1)| ≤ 1. A graph G is cordial if it admitscordial labeling.

The concept of cordial labeling was introduced by Cahit[4].

Many researchers have studied cordiality of graphs.e.g.Cahit [4] proved that tree is cordial. In the same paperhe proved that Kn is cordial if and only if n ≤ 3. Ho et al.[9] proved that unicyclic graph is cordial unless it is C4k+2

while Andar et al. [1] have discussed cordiality of multipleshells. Vaidya et al. [10], [11], [12] have also discussed thecordiality of various graphs.

Definition 1.6 A vertex labeling of a graph G iscalled a 3-equitable labeling if |vf (i) − vf (j)| ≤ 1and |ef (i) − ef (j)| ≤ 1 for all 0 ≤ i, j ≤ 2. A graph G is3-equitable if it admits 3-equitable labeling.

The concept of 3-equitable labeling was introduced byCahit [5] and in the same paper he proved that Euleriangraphs with number of edges congruent to 3(mod6) are not3-equitable. Youssef [17] proved that Wn is 3-equitable forall n ≥ 4. Several results on 3-equitable labeling for somewheel related graphs in the context of vertex duplication arereported in Vaidya et al. [13].

In the present investigations we prove that graphs< W

(1)n : W

(2)n > and < W

(1)n : W

(2)n : W

(3)n : . . . : W

(k)n >

are cordial as well as 3-equitable.

II. MAIN RESULTS

Theorem-2.1 Graph < W(1)n : W

(2)n > is cordial.

Proof Let v(1)1 , v

(1)2 , v

(1)3 , . . . v

(1)n be the rim vertices

W(1)n and v

(2)1 , v

(2)2 , v

(2)3 , . . . v

(2)n be the rim vertices W

(2)n .

Let c1 and c2 be the apex vertices of W(1)n and W

(2)n

IAENG International Journal of Applied Mathematics, 41:2, IJAM_41_2_03

(Advance online publication: 24 May 2011)

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respectively and they are adjacent to a new common vertexx. Let G =< W

(1)n : W

(2)n >. We define binary vertex

labeling f : V (G) → {0, 1} as follows.

For any n ∈ N − {1, 2} and i = 1, 2, . . . n where Nis set of natural numbers.In this case we define labeling as follows

f(v(1)i ) = 1;

f(c1) = 0;f(v

(2)i ) = 0;

f(c2) = 1;f(x) = 1;

Thus rim vertices of W(1)n and W

(2)n are labeled with

the sequences 1, 1, 1, . . . , 1 and 0, 0, . . . , 0 respectively. Thecommon vertex x is labeled with 1 and apex vertices with0 and 1 respectively.

The labeling pattern defined above covers all possiblearrangement of vertices. The graph G satisfies thevertex condition vf (0) + 1 = vf (1) and edge conditionef (0) = ef (1). i.e. G admits cordial labeling.

Illustration 2.2 Consider G =< W(1)6 : W

(2)6 >.

Here n = 6. The cordial labeling is as shown in Figure 1.

Theorem 2.3 Graph < W(1)n : W

(2)n : W

(3)n : . . . : W

(k)n >

is cordial.

Proof Let W(j)n be k copies of wheel Wn, v

(j)i be

the rim vertices of W(j)n and cj be the apex vertex

of W(j)n (here i = 1, 2, . . . n and j = 1, 2, . . . k).Let

x1, x2 . . . xk−1 be the vertices such that cp−1 andcp are adjacent to xp−1 where 2 ≤ p ≤ k. ConsiderG =< W

(1)n : W

(2)n : W

(3)n : . . . : W

(k)n >. To define binary

vertex labeling f : V (G) → {0, 1} we consider followingcases.

Case 1: n ∈ N − {1, 2} and even k where k ∈ N − {1, 2}.In this case we define labeling function f asFor i = 1, 2, . . . n and j = 1, 2, . . . kf(v

(j)i ) = 0; if j even.

= 1; if j odd.f(cj) = 1; if j even.

= 0; if j odd.f(xj) = 1; if j even, j 6= k.

= 0; if j odd, j 6= k.Case 2: n ∈ N − {1, 2} and odd k where k ∈ N − {1, 2}.In this case we define labeling function f for first k − 1wheels as

Fig. 1. Cordial labeling of graph G.

For i = 1, 2, . . . n and j = 1, 2, . . . k − 1f(v

(j)i ) = 0; if j even.

= 1; if j odd.f(cj) = 1; if j even.

= 0; if j odd.f(xj) = 1; if j even.

= 0; if j odd.To define labeling function f for kth copy of wheel weconsider following subcases

Subcase 1: If n ≡ 3(mod4).For 1 ≤ i ≤ n− 1f(v

(k)i ) = 0; if i ≡ 0, 1(mod4).

= 1; if i ≡ 2, 3(mod4).f(v

(k)n ) = 0;

f(ck) = 1;

Subcase 2: If n ≡ 0, 2(mod4).f(v

(k)i ) = 0; if i ≡ 0, 1(mod4).

= 1; if i ≡ 2, 3(mod4).f(ck) = 0; n ≡ 0(mod4)f(ck) = 1; n ≡ 2(mod4)

Subcase 3: If n ≡ 1(mod4).f(v

(k)i ) = 0; if i ≡ 0, 3(mod4).

= 1; if i ≡ 1, 2(mod4).f(ck) = 0;

The labeling pattern defined above exhaust allthe possibilities and in each one the graph G underconsideration satisfies the conditions |vf (0) − vf (1)| ≤ 1and |ef (0)− ef (1)| ≤ 1 as shown in Table 1. i.e. G admitscordial labeling.

(In Table 1 n = 4a+ b and a ∈ N ∪ {0})Let us understand the labeling pattern with some examples

given below.

Illustrations 2.4Example 1: Consider G =< W

(1)7 : W

(2)7 : W

(3)7 :

W(4)7 >. Here n = 7 and k = 4 i.e k is even. The cordial

labeling is as shown in Figure 2.

Example 2: Consider G =< W(1)5 : W

(2)5 : W

(3)5 >. Here

n = 5 i.e n ≡ 1(mod4) and k = 3 i.e k is odd. The cordiallabeling is as shown in Figure 3.


(2)n > is 3-equitable.

Proof Let v(1)1 , v

(1)2 , v

(1)3 , . . . v

(1)n be the rim vertices

W(1)n and v

(2)1 , v

(2)2 , v

(2)3 , . . . v

(2)n be the rim vertices W

(2)n .

TABLE IVERTEX AND EDGE CONDITIONS FOR f



______________________________________________________________________________________



Let c1 and c2 be the apex vertices of W(1)n and W

(2)n

respectively and they are adjacent to a new common vertexx. Let G =< W

(1)n : W

(2)n >. To define vertex labeling

f : V (G) → {0, 1, 2} we consider the following cases.

Case 1: n ≡ 0(mod6)In this case we define labeling f as:f(v

(1)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 2, 3(mod6)= 2; i ≡ 0, 5(mod6), 1 ≤ i ≤ n

f(c1) = 2;f(v

(2)i ) = 0; i ≡ 1, 4(mod6)

= 2; i ≡ 2, 3(mod6)= 1; i ≡ 0, 5(mod6), 1 ≤ i ≤ n− 3= 1; i ≥ n− 2

f(c2) = 0;f(x) = 0;

Case 2: n ≡ 1(mod6)In this case we define labeling f as:f(v

(1)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 2, 3(mod6)= 2; i ≡ 0, 5(mod6), 1 ≤ i ≤ n

f(c1) = 2;f(v

(2)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 2, 3(mod6)= 2; i ≡ 0, 5(mod6), 1 ≤ i ≤ n

f(c2) = 2;f(x) = 1;

Case 3: n ≡ 2(mod6)In this case we define labeling f as:

f(v(1)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 0, 5(mod6)= 2; i ≡ 2, 3(mod6), 1 ≤ i ≤ n− 2= 1; i ≥ n− 1

f(c1) = 0;f(v

(2)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 0, 5(mod6)= 2; i ≡ 2, 3(mod6), 1 ≤ i ≤ n− 2

= 2; i ≥ n− 1f(c2) = 0;f(x) = 1;

Case 4: n ≡ 3(mod6)Subcase 1: n 6= 3In this case we define labeling f as:

f(v(1)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 0, 5(mod6)= 2; i ≡ 2, 3(mod6), 1 ≤ i ≤ n

f(c1) = 0;f(v

(2)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 2, 3(mod6)= 2; i ≡ 0, 5(mod6), 1 ≤ i ≤ n− 3= 1; i ≥ n− 2

f(c2) = 0;f(x) = 2;

Subcase 2: n = 3f(v

(1)1 ) = f(v

(2)1 ) = f(c2) = 0;

f(v(1)2 ) = f(v

(1)3 ) = f(c1) = 1;

f(v(2)2 ) = f(v

(2)3 ) = f(x) = 2;

Case 5: n ≡ 4(mod6)In this case we define labeling f as:

f(v(1)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 0, 5(mod6)= 2; i ≡ 2, 3(mod6), 1 ≤ i ≤ n− 3= 1; i = n− 2, n− 1= 0; i = n

f(c1) = 2;f(v

(2)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 0, 5(mod6)= 2; i ≡ 2, 3(mod6), 1 ≤ i ≤ n

f(c2) = 2; f(x) = 1.Case 6: n ≡ 5(mod6)In this case we define labeling f as:

f(v(1)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 2, 3(mod6)= 2; i ≡ 0, 5(mod6), 1 ≤ i ≤ n− 5= 1; i = n− 4, n− 3= 2; i = n− 2, n= 0; i = n− 1

f(c1) = 2;f(v

(2)i ) = 0; i ≡ 1, 4(mod6)

= 1; i ≡ 0, 5(mod6)= 2; i ≡ 2, 3(mod6), 1 ≤ i ≤ n− 5= 0; i = n− 4, n− 1



______________________________________________________________________________________

= 1; i = n− 3, n− 2= 2; i = n

f(c2) = 0;f(x) = 0;

The labeling pattern defined above covers all thepossible arrangement of vertices and in each case theresulting labeling satisfies the conditions |vf (i)−vf (j)| ≤ 1and |ef (i) − ef (j)| ≤ 1 for all 0 ≤ i, j ≤ 2 as shown inTable 2. i.e. G admits 3-equitable labeling.

(In Table 2 n = 6a+ b and a ∈ N ∪ {0})Let us understand the labeling pattern defined in Theorem2.5 by means of following Illustration 2.6.

Illustration 2.6 Consider a graph G =< W(1)5 : W

(2)5 >

Here n = 5 i.e n ≡ 5(mod6). The corresponding 3-equitablelabeling is shown in Figure 4.


(2)n : W

(3)n : . . . : W

(k)n >

is 3-equitable.

Proof Let W(j)n be k copies of wheel Wn, v

(j)i be

the rim vertices of W(j)n where i = 1, 2, . . . n and

j = 1, 2, . . . k. Let cj be the apex vertex of W(j)n . Consider

G =< W(1)n : W

(2)n : W

(3)n : . . . : W

(k)n > and vertices

x1, x2, . . . xk−1 as stated in Theorem 2.3. To define vertexlabeling f : V (G) → {0, 1, 2} we consider following cases.

Case 1: For n ≡ 0(mod6).In this case we define labeling function f as followsSubcase 1: For k ≡ 0(mod3).

For j ≡ 1, 2(mod3)

f(v(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 3.

f(v(j)i ) = 1; if i ≥ n− 2.

f(cj) = 0.f(xj) = 2; if j ≡ 1(mod3).

= 0; if j ≡ 2(mod3).For j ≡ 0(mod3)

f(v(j)i ) = 0; if i ≡ 1, 4(mod6).

TABLE IIVERTEX AND EDGE CONDITIONS FOR f

Fig. 4. 3-equitable labeling of graph G.

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6).

f(cj) = 2.f(xj) = 0, j 6= k.

Subcase 2: For k ≡ 1(mod3).f(v

(1)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6).

f(c1) = 2.f(x1) = 0.

For remaining vertices take j = k − 1 and label them asin subcase 1.Subcase 3: For k ≡ 2(mod3).f(v

(1)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6).

f(c1) = 0.f(x1) = 2.

f(v(2)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 3.

f(v(2)i ) = 1; if i ≥ n− 2.

f(c2) = 0.f(x2) = 0.

For remaining vertices take j = k − 2 and label them asin subcase 1.Case 2: For n ≡ 1(mod6).In this case we define labeling function f as followsSubcase 1: For k ≡ 0(mod3).f(v

(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 1.

f(v(j)n ) = 0; if j ≡ 1(mod3).

f(v(j)n ) = 1; if j ≡ 0, 2(mod3).

f(cj) = 2; if j ≡ 1(mod3).f(cj) = 0; if j ≡ 0, 2(mod3).f(xj) = 1; if j ≡ 1(mod3).

= 2; if j ≡ 0, 2(mod3), j 6= k.Subcase 2: For k ≡ 1(mod3).f(v

(1)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 1.

f(v(1)n ) = 1;

f(c1) = 2.f(x1) = 0.

For remaining vertices take j = k − 1 and label them asin subcase 1.Subcase 3: For k ≡ 2(mod3).For j = 1, 2f(v

(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 1.

f(v(j)n ) = 1;

f(c1) = 0.f(c2) = 2.f(x1) = 2.f(x2) = 0.

For remaining vertices take j = k − 2 and label them asin subcase 1.



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For j ≡ 1, 2(mod3)

f(v(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 4.

f(v(j)n−3) = 2.f(v

(j)i ) = 1; if i ≥ n− 2.

f(cj) = 0; if j ≡ 1(mod3).f(cj) = 2; if j ≡ 2(mod3).f(xj) = 0.

For j ≡ 0(mod3)

f(v(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 2, 3(mod6).= 2; if i ≡ 0, 5(mod6), i ≤ n− 2.

f(v(j)i ) = 1; if i ≥ n− 1.

f(cj) = 2.f(xj) = 0, j 6= k.


(1)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 2.

f(v(1)n−1) = 2.f(v

(1)n ) = 0.

f(c1) = 0.f(x1) = 1.


(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 4.

f(v(j)n−3) = 2;f(v

(j)i ) = 1; if i ≥ n− 2.

f(cj) = 0.f(x1) = 1.f(x2) = 0.

For remaining vertices take j = k − 2 and label them asin subcase 1.Case 4: For n ≡ 3(mod6).In this case we define labeling function f as followsSubcase 1: For k ≡ 0(mod3).

f(v(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 3.

If j ≡ 1(mod3)

f(v(j)i ) = 1; if i ≥ n− 2.

f(cj) = 0.f(xj) = 1.

If j ≡ 2(mod3)

f(v(j)n−2) = 0.

f(v(j)n−1) = 2.f(v

(j)n ) = 1.

f(cj) = 0.f(xj) = 2.

If j ≡ 0(mod3)

f(v(j)i ) = 0; ifj = n− 1, n− 2.

f(v(j)n ) = 2.

f(cj) = 2.f(xj) = 2, j 6= k.


(1)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 2, 3(mod6).= 2; if i ≡ 0, 5(mod6), i ≤ n− 3.

f(v(1)i ) = 2; if i ≥ n− 2.

f(c1) = 0.f(x1) = 1.


(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 3.

f(v(1)i ) = 1; if i = n− 1, n− 2.

f(v(1)n ) = 0.

f(v(2)i ) = 2; if i ≥ n− 2.

f(cj) = 0.f(x1) = 1.f(x2) = 2.

For n = 3 label rim vertices of W(1)n by 0, 1, 0 and apex

vertex by 1.For remaining vertices take j = k − 2 and label them as

in subcase 1.Case 5: For n ≡ 4(mod6).In this case we define labeling function f as followsSubcase 1: For k ≡ 0(mod3).

For j ≡ 0, 1, 2(mod3)

f(v(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 4.

f(v(j)n−3) = 0; if j ≡ 0, 1(mod3).

f(v(j)n−3) = 2; if j ≡ 2(mod3).f(v

(j)i ) = 1; if j ≡ 1, 2(mod3), i ≥ n− 2.

f(v(j)i ) = 2; if j ≡ 0(mod3), i ≥ n− 2.

f(cj) = 2, j ≡ 1, 2(mod3).f(cj) = 0, j ≡ 0(mod3).f(xj) = 0, j 6= k.


(1)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6).

f(c1) = 0.f(x1) = 1.

For remaining vertices take j = k − 1 and label them asin subcase 1.Subcase 3: For k ≡ 2(mod3).f(v

(1)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 2, 3(mod6).= 2; if i ≡ 0, 5(mod6).

f(v(2)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6).

f(c1) = 2.



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f(c2) = 0.f(x1) = 1.f(x2) = 2.



For j ≡ 1, 2(mod3)

f(v(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 2, 3(mod6).= 2; if i ≡ 0, 5(mod6), i ≤ n− 2.

f(v(j)n−1) = 1.f(v

(j)n ) = 2; if j ≡ 1(mod3).

f(v(j)n ) = 0; if j ≡ 2(mod3).

f(cj) = 2; if j ≡ 1(mod3).f(cj) = 0; if j ≡ 2(mod3).f(xj) = 1; if j ≡ 1(mod3).f(xj) = 2; if j ≡ 2(mod3).

For j ≡ 0(mod3)

f(v(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6) i ≤ n− 1.

f(v(j)n ) = 2.

f(cj) = 0.f(xj) = 2, j 6= k.


(1)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 2.

f(v(1)i ) = 1; if i ≥ n− 1.

f(c1) = 0.f(x1) = 2.


(j)i ) = 0; if i ≡ 1, 4(mod6).

= 1; if i ≡ 0, 5(mod6).= 2; if i ≡ 2, 3(mod6), i ≤ n− 2.

f(v(j)i ) = 1, i ≥ n− 1.

f(c1) = 0.f(c2) = 2.f(xj) = 0.


The labeling pattern defined above covers all possiblearrangement of vertices. In each case, the graph G underconsideration satisfies the conditions |vf (i) − vf (j)| ≤ 1and |ef (i) − ef (j)| ≤ 1 for all 0 ≤ i, j ≤ 2 as shown inTable 3. i.e. G admits 3-equitable labeling.

(In Table 3 n = 6a + b and k = 3c + d wherea ∈ N ∪ {0},c ∈ N )

The labeling pattern defined above is demonstrated by meansof following Illustration 2.8.

Illustration 2.8 Consider a graph G =< W(1)6 : W

(2)6 :

W(3)6 : W

(4)6 >. Here n = 6 and k = 4. The corresponding

3-equitable labeling is as shown in Figure 5.

TABLE IIIVERTEX AND EDGE CONDITIONS FOR f

III. CONCLUDING REMARKS

Cordial and 3-equitable labeling of some star and shellrelated graphs are reported in Vaidya et al. [14], [15] whilethe present work corresponds to cordial and 3-equitablelabeling of some wheel related graphs. Here we providecordial and 3-equitable labeling for the larger graphsconstructed from the standard graph.

Further scope of research

• Similar investigations can be carried out in the contextof different graph labeling techniques and for variousstandard graphs.

• Cordial labeling in the context of arbitrary super subdi-vision of graphs is discussed in Vaidya and Kanani [16].In likeway all the results reported here can be discussedin the context of arbitrary super subdivision of graphs.

REFERENCES

[1] M Andar, S Boxwala and N B Limaye: “A Note on cordial labelingof multiple shells”, Trends Math., pp. 77-80, 2002.

[2] L W Beineke and S M Hegde, “Strongly Multiplicative graphs”,Discuss.Math. Graph Theory,21, pp.63-75, 2001.(http://www.discuss.wmie.uz.zgora.pl/gt/index.php)

[3] G S Bloom and S W Golomb, “Application of numbered undirectedgraphs”, Proceedings of IEEE, 65(4), pp. 562-570, 1977.(http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=31252)

[4] I Cahit, Cordial Graphs: “A weaker version of graceful and harmoniousGraphs”, Ars Combinatoria, 23, pp. 201-207, 1987.

[5] I Cahit, “On cordial and 3-equitable labelings of graphs”, Util. Math.,37, pp. 189-198, 1990.

[6] J A Gallian, A dynamic survey of graph labeling, The ElectronicsJournal of Combinatorics, 17 ]DS6, 2010.(http://www.combinatorics.org/Surveys/index.html)

[7] F Harary, Graph theory, Addison Wesley, Reading, Massachusetts,1972.

[8] S M Hegde, “On Multiplicative Labelings of a Graph”, Labeling ofDiscrete Structures and applications, Narosa Publishing House, NewDelhi, pp. 83-96, 2008.



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Fig. 5. 3-equitable labeling of graph G.

[9] Y S Ho, S M Lee and S C Shee, “Cordial labeling of unicyclic graphsand generalized Petersen graphs”, Congress. Numer.,68, pp. 109-122,1989.

[10] S K Vaidya, G V Ghodasara, Sweta Srivastav, V J Kaneria, “Cordiallabeling for two cycle related graphs”, The Mathematics Student, J. ofIndian Mathematical Society. 76, pp. 237-246, 2007.

[11] S K Vaidya, G V Ghodasara, Sweta Srivastav, V J Kaneria, “Somenew cordial graphs”, Int. J. of scientific comp. 2(1), pp. 81-92, 2008.

[12] S K Vaidya, Sweta Srivastav, G V Ghodasara, V J Kaneria, “Cordiallabeling for cycle with one chord and its related graphs”, Indian J. ofMath. and Math.Sci 4(2), pp. 145-156, 2008.

[13] S K Vaidya, N A Dani, K K Kanani, P L Vihol, “Some wheel related 3-Equitable Graphs in the context of vertex duplication”, Advance Appl.in Discrete Math. 4(1), pp. 71-85, 2009.(www.pphmj.com/journals/aadm.html)

[14] S K Vaidya, N A Dani, K K Kanani, P L Vihol, “Cordial and 3-Equitable labeling for some star related graphs”, Int. Math. Forum4(31), pp. 1543-1553, 2009.(www.m-hikari.com/imf.html)

[15] S K Vaidya, N A Dani, K K Kanani, P L Vihol, “Cordial and 3-Equitable labeling for some shell related graphs”, J. Sci. Res. 1(3),pp. 438-449, 2009.(www.banglajol.info/index.php/JSR)

[16] S K Vaidya, K K Kanani, “Some new results on cordial labeling in thecontext of abritray supersubdivision of graph”, Applied Math. Sciences4(47), pp. 2323-2329, 2010.(www.m-hikari.com/ams/index.html)

[17] M. Z. Youssef, “A necessary condition on k-equitable labelings”, Util.Math., 64, pp. 193-195, 2003.



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