4-Difference Cordial Labeling of Cycle andWheel Related Graphs

S. M. Vaghasiya1, G. V. Ghodasara2

1 Research Scholar, R. K. University, Rajkot−360020, India.2 H. & H. B. Kotak Institute of Science,Rajkot−360001, India.

1 sarla.spkm@gmail.com

2 gaurang enjoy@yahoo.co.in

Abstact —Let G be a (p, q) graph. Letk be an integer with 2 ≤ k ≤ p andf : V (G)→ {1, 2, . . . k} be a map. For each edgeuv, assign the label |f(u)− f(v)|. The functionf is called a k-difference cordial labeling of Gif |vf (0) − vf (1)| ≤ 1 and |ef (0) − ef (1)| ≤ 1where vf (x) denotes the number of verticeslabelled with x (x ∈ {1, 2, . . . , k}), ef (1) andef (0) respectively denote the number of edgeslabelled with 1 and not labelled with 1. Agraph with a k-difference cordial labeling iscalled a k-difference cordial graph. In thispaper we discuss 4-difference cordial labelingfor cycle, wheel, crown, helm and gear graph.

Key words : Difference cordial labeling, 4-difference cordial labeling.Subject classification number: 05C78.

I. INTRODUCTION

We consider simple, finite, undirectedgraph G = (V,E). R. Ponraj, M. MariaAdaickalam and R. Kala [6] introduced k-difference cordial labeling of graphs. In[6], they investigated k-difference cordiallabeling behavior of star, m copies of starand proved that every graph is a subgraphof a connected k-difference cordial graph.In [7], R. Ponraj and M. Maria Adaickalamdiscussed the 3-difference cordial label-ing behavior of path, cycle, star, bistar,complete graph, complete bipartite graph,comb, double comb, quadrilateral snake.For the standard terminology and notationswe follow Harary[1].

II. MAIN RESULTS

In this paper we have proved that cycle,wheel, helm, crown and gear graph are 4-difference cordial graphs.

Definition II.1. A cycle Cn(n ∈ N, n ≥ 3)is closed path with n vertices.

Theorem II.1. Cycle Cn is a 4-differencecordial graph.

Proof. Let V (Cn) = {v1, v2, . . . , vn}.We define labeling functionf : V (Cn)→ {1, 2, 3, 4} as follows.Case 1: n is odd.

f(v4i+1) = 1; 0 ≤ i ≤ bn− 1

4c.

f(v4i+2) = 2; 0 ≤ i ≤ bn− 3

4c.

f(v4i) = 3; 1 ≤ i ≤ bn− 1

4c.

f(v4i+3) = 4; 0 ≤ i ≤ bn− 3

4c.

Case 2: n is even.Subcase 1: n ≡ 0(mod4).

f(v4i) = 1; 1 ≤ i ≤ n

4.

f(v4i+3) = 2; 0 ≤ i ≤ n− 4

4.

f(v4i+1) = 3; 0 ≤ i ≤ n− 4

4.

f(v4i+2) = 4; 0 ≤ i ≤ n− 4

4.

Subcase 2: n ≡ 2(mod4).

f(v1) = 2,

f(v2) = 1.

f(v4i+1) = 1; 1 ≤ i ≤ n− 2

4.

f(v4i+2) = 2; 1 ≤ i ≤ n− 2

4.

f(v4i) = 3; 1 ≤ i ≤ n− 2

4.

f(v4i+3) = 4; 0 ≤ i ≤ n− 6

4.

In each case cycle Cn satisfies the condi-tions for 4-difference cordial labeling.Hence Cn is a 4-difference cordialgraph.

Example 1. The 4-difference cordial la-beling of C18 is shown in Figure 1.

1

1

1

1

1

2

2

22

2

3 3

3

4

4

4

4

3

Fig. 1

Definition II.2. The wheel Wn(n ∈N, n ≥ 3) is a join of the graphs Cn andK1. i.e Wn = Cn +K1.Here vertices corresponding to Cn arecalled rim vertices and Cn is called rimof Wn.The vertex corresponding to K1 is calledapex vertex.

Theorem II.2. Wn is a 4-difference cor-dial graph.

Proof. Let v0 be the apex vertex and

v1, v2, . . . , vn be the rim vertices of Wn.We define labeling function f : V (Wn)→{1, 2, 3, 4} as follows.Case 1: n is odd.

f(v4i) = 1 ; 1 ≤ i ≤ bn− 1

4c.

f(v4i+1) = 2 ; 0 ≤ i ≤ bn− 1

4c.

f(v4i+2) = 3 ; 0 ≤ i ≤ bn− 3

4c.

f(v4i+3) = 4 ; 0 ≤ i ≤ bn− 3

4c.

Case 2: n is even.

f(v1) = 2,

f(v2) = 3,

f(v3) = 4.

f(v4i+3) = 1; 1 ≤ i ≤ bn− 3

4c.

f(v4i) = 2; 1 ≤ i ≤ bn4c.

f(v4i+1) = 3; 1 ≤ i ≤ bn− 2

4c.

f(v4i+2) = 4; 1 ≤ i ≤ bn− 2

4c.

In each case wheel graph Wn satisfies theconditions of 4-difference cordial labeling.Hence Wn is 4-difference cordial graph.

Example 2. 4-difference cordial labelingof W11 is shown in Figure 2.

1

1

1

22

2

3

3

3

4

4

4

Fig. 2

Definition II.3. [3] The crown Cn �K1(n ∈ N, n ≥ 3) is obtained by joininga pendant edge to each vertex of Cn.

Theorem II.3. Crown Cn � K1 is a 4-difference cordial graph.

Proof. Let V (Cn � K1) ={v1, v2, . . . , vn, v′1, v′2, . . . , v′n}, wherev1, v2, . . . , vn are rim vertices andv′1, v

′2, . . . , v

′n are pendant vertices.

We define labeling functionf : V (Cn � K1) → {1, 2, 3, 4} asfollows.Case 1: n is odd.

f(v2i+1) = 1; 0 ≤ i ≤ n− 1

2.

f(v2i) = 3; 1 ≤ i ≤ n− 1

2.

f(v′2i+1) = 2; 0 ≤ i ≤ n− 1

2.

f(v′2i) = 4; 1 ≤ i ≤ n− 1

2.

Case 2: n is even.

f(v2i+1) = 1; 0 ≤ i ≤ n− 2

2.

f(v2i) = 3; 1 ≤ i ≤ n

2.

f(v′2i+1) = 2; 0 ≤ i ≤ n− 2

2.

f(v′2i) = 4; 1 ≤ i ≤ n

2.

In each case the crown graph Cn�K1 sat-isfies the conditions of 4-difference cordiallabeling. Hence it is 4-difference cordialgraph.

Example 3. 4-difference cordial labelingof crown C9 �K1 is shown in Figure 3.

1

1

1

1

1

3

3

3

3

2

2

2

2

2

4

4

4

4

Fig. 3

Definition II.4. A helm Hn(n ≥ 3) isthe graph obtained from the wheel Wn byadding a pendant edge at each vertex onthe rim of Wn.

Theorem II.4. Hn is a 4-difference cor-dial graph.

Proof. Let V (Hn) ={v0, v1, . . . , vn, v′1, v′2, . . . , v′n}, wherev0 is apex vertex, {v1, v2, . . . , vn} are rimvertices and {v′1, v′2, . . . , v′n} are pendantvertices.We define labeling functionf : V (Hn)→ {1, 2, 3, 4} as follows.Case 1: n is odd.

f(v4i) = 1; 1 ≤ i ≤ bn− 1

2c.

f(v4i+1) = 2; 0 ≤ i ≤ bn− 1

2c.

f(v4i+2) = 3; 0 ≤ i ≤ bn− 3

4c.

f(v4i+3) = 4; 0 ≤ i ≤ bn− 3

4c.

f(v′4i+3) = 1; 0 ≤ i ≤ bn− 3

4c.

f(v′4i+2) = 2; 0 ≤ i ≤ bn− 3

4c.

f(v′4i+1) = 3; 0 ≤ i ≤ bn− 1

4c.

f(v′4i) = 4; 1 ≤ i ≤ bn− 1

4c.

Case 2: n is even.

f(v2i+1) = 2; 0 ≤ i ≤ n− 2

2.

f(v2i) = 4; 1 ≤ i ≤ n

2.

f(v′2i+1) = 1; 0 ≤ i ≤ n− 2

2.

f(v′2i) = 3; 1 ≤ i ≤ n

2.

In each case the helm graph Hn satisfiesthe conditions of 4-difference cordial la-beling. Hence Hn is 4-difference cordialgraph.

Example 4. 4-difference cordial labelingof helm H9 is shown in Figure 4.

2

4

1

2

4

3

3

2

1

1

3

1

3

4

4

3

2

2

1

Fig. 4

Definition II.5. A gear graph Gn(n ≥ 3)is obtained from the wheel Wn by addinga vertex between every pair of adjacentvertices of rim of Wn.

Theorem II.5. Gear Gn is a 4-differencecordial graph.

Proof. Let Gn={v0, v1, . . . , v2n}, where v0is apex vertex, {v1, v3, . . . , v2n−1} are thevertices of degree 3 and {v2, v4, . . . , v2n}are the vertices of degree 2.We define labeling function f : V (Gn)→{1, 2, 3, 4} as follows.

Case 1: n is odd.

v0 = 3.

f(v4i+1) = 1; 0 ≤ i ≤ n− 1

2.

f(v4i+2) = 2; 0 ≤ i ≤ n− 1

2.

f(v4i+3) = 3; 0 ≤ i ≤ n− 3

2.

f(v4i+4) = 4; 0 ≤ i ≤ n− 3

2.

Case 2: n is even.

v0 = 1.

f(v4i+1) = 1; 0 ≤ i ≤ n

2− 1.

f(v4i+2) = 2; 0 ≤ i ≤ n

2− 1.

f(v4i+3) = 3; 0 ≤ i ≤ n

2− 1.

f(v4i+4) = 4; 0 ≤ i ≤ n

2− 1.

In each case the gear graph Gn satisfiesthe conditions of 4-difference cordial la-beling. Hence Gn is 4-difference cordialgraph.

Example 5. 4-difference cordial labelingof G5 is shown in Figure 5.

1

4

1

31

2

3

3

1

4

2

Fig. 5

REFERENCES

[1] F. Harary, Graph theory, Addision-wesley, Reading,MA (1969).

[2] I. Cahit, On cordial and 3−equitable labelings ofgraphs, Util. Math., 37(1990), 189-198.

[3] J. A. Gallian, A dynemic survey of graphlabeling, The Electronics Journal of Combinatorics,16(2013), ]DS6 1− 308.

[4] J. Gross and J. Yellen, Graph theory and itsapplications, CRC Press, (1999).

[5] R. Ponraj, S. Sathish Narayanan and R. Kala,Difference Cordial Labeling of Graphs, GlobalJournal of Mathematical Sciences: Theory andPractical, 5 (2013) 185-196.

[6] R. Ponraj, M. Maria Adaickalam and R. Kala, k-difference cordial labeling of graphs, InternationalJournal of Mathematical Combinatorics, 2 (2016),121-131.

[7] R. Ponraj and M. Maria Adaickalam, 3-differencecordial labeling of some cycle related graphs,Journal of Algorithms and Computation, 47 (2016),1-10.

[8] S. M. Vaghasiya and G. V. Ghodasara, DifferenceCordial of Operational Graph Related to Cycle,International Journal of Advanced EngineeringResearch and Science, 3 (2016), 236-239.