Difference Cordial Labeling of Subdivision of Snake Graphs

Difference Cordial Labeling of Subdivision of

Snake Graphs

R. Ponraj

1,∗

_,

_{S. Sathish Narayanan}

2

_,

_{R. Kala}

3

1_{Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-627412, India}

2_{Department of Mathematics, Thiruvalluvar College, Papanasam-627425, India}

3_{Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, India}

∗_{Corresponding Author: ponrajmaths@gmail.com}

Copyright c⃝2014 Horizon Research Publishing All rights reserved.

Abstract

Let G be a (p, q) graph. Let f : V (G) → {1,2. . . , p} be a function. For each edge uv, assign the label |f(u)−f(v)|. f is called a difference cordial labeling iff is a one to one map and

|ef(0)−ef(1)| ≤ 1 where ef(1) and ef(0) denote the

number of edges labeled with 1 and not labeled with 1 respectively. A graph with a difference cordial labeling is called a difference cordial graph. In this paper, we investigate the difference cordial labeling behavior of subdivision of some snake graphs.

Keywords

Triangular snake, Quadrilateral snake, Subdivision, Double triangular snake, Double quadri-lateral snake

1

Introduction

Let G = (V, E) be a (p, q) graph. In this paper we have considered only simple, non-trivial and undirected graphs. The number of vertices ofGis called the order ofGand the number of edges ofGis called the sizeG. The subdivision graphS(G) of a graphGis obtained by replacing each edgeuv by a pathuwv. Graph labeling plays a vital role of numerous fields of sciences and some of them are astronomy, coding theory, x-ray crystallog-raphy, radar, database management etc.[1]. Ponrajet al. introduced difference cordial labeling of grpahs [3]. Let Gbe a (p, q) graph. Letf : V(G)→ {1,2. . . , p} be a map. For each edgeuv, assign the label|f(u)−f(v)|. f is called a difference cordial labeling if f is a one to one map and |ef(0)−ef(1)| ≤ 1 where ef(1) and

ef(0) denote the number of edges labeled with 1 and

not labeled with 1 respectively. A graph with admits difference cordial labeling is called a difference cordial graph. In [3, 4, 5, 6] difference cordial labeling behav-ior of several graphs like path, cycle, complete graph, complete bipartite graph, bistar, wheel, web,Wn⊙K2,

Pm⊙Pn and some more standard graphs have been

in-vestigated. In this paper, we investigate the difference cordial behaviour of S(Tn), S(A(Tn)) S(DA(Tn)),

S(Qn), S(A(Qn)), S(DA(Qn)). Terms not defined

here are used in the sense of Harary [2].

2

Methodology

In this article we use some number theory techniques for proving the main results.

3

Main Results

The triangular snake Tn is obtained from the path

Pn by replacing each edge of the path by a triangleC3.

S(T2)∼=C6 and hence difference cordial [3].

Theorem 3.1 S(Tn)is difference cordial for alln >2.

Proof. Let Pn be the path u1u2. . . un. Let

V (Tn) =V (Pn)∪{vi: 1≤i≤n−1}. LetV(S(Tn)) =

{xi, yi, wi : 1≤i≤n−1} ∪ V(Tn) and E(S(Tn)) =

{uixi, xivi, yivi, yiui+1, uiwi, wiui+1: 1≤i≤n−1}.

Define an injective map f : V(S(Tn)) →

{1,2, . . . ,5n−4}by

f(ui) = 2i−1 1≤i≤n

f(wi) = 2i 1≤i≤n−1

f(xi) = 2n+ 2i−2 1≤i≤n−1

f(yi) = 4n−3 +i 1≤i≤n−1

f(vi) = 2n+ 2i−1 1≤i≤n−1.

Since ef(0) =ef(1) = 3n−3,f is a difference cordial

labeling ofS(Tn).

The Quadrilateral snakeQnis obtained from the path

Pn by replacing each edge of the path by a cycleC4.

Theorem 3.2 S(Qn)is difference cordial.

Proof. LetPnbe the pathu1u2. . . unand LetV (Qn) =

{vi, wi: 1≤i≤n−1} ∪ V(Pn). Let V(S(Qn)) =

{

xi, u

′

i, zi, yi: 1≤i≤n−1

}

∪ V (Qn) and

E(S(Qn)) =

{

uiu

′

i, u

′

iui+1, yiui+1: 1≤i≤n−1

{uixi, xivi, vizi, ziwi, wiyi: 1≤i≤n−1}. Define a

mapf :V(S(Qn))→ {1,2, . . . ,7n−6}by

f(ui) = 2i−1 1≤i≤n

f

(

u′_i

)

= 2i 1≤i≤n−1

f(vi) = 2n+ 3i−3 1≤i≤n−1

f(zi) = 2n+ 3i−2 1≤i≤n−1

f(wi) = 2n+ 3i−1 1≤i≤n−1

f(xi) = 5n−4 +i 1≤i≤n−1

f(yi) = 6n−5 +i 1≤i≤n−1.

Since ef(0) =ef(1) = 4n−4,f is a difference cordial

labeling ofS(Qn).

A double triangular snake DTn consists of two

trian-gular snakes that have a common path.

Theorem 3.3 S(DTn) is difference cordial.

Proof. Let V(S(DTn)) = {ui: 1≤i≤n} ∪

{

xi, yi, vi, x

′

i, y

′

i, wi, zi: 1≤i≤n−1

}

andE(S(DTn))

=

{

uizi, ziui+1, uixi, yiui+1, xivi, uix

′

i: 1≤i≤n−1

} ∪ {

viyi, wiy

′

i, x

′

iwi, y

′

iui+1: 1≤i≤n−1

}

. Define a map f :V (S(DTn))→ {1,2, . . . ,8n−7} by

f(ui) = 4i−3 1≤i≤n

f(zi) = 7n−6 +i 1≤i≤n−1

f(xi) = 4i−2 1≤i≤n−1

f(vi) = 4i−1 1≤i≤n−1

f(yi) = 4i 1≤i≤n−1

f

(

x′_i

)

= 4n+ 2i−4 1≤i≤n−1 f(wi) = 4n+ 2i−3 1≤i≤n−1

f

(

y_i′

)

= 6n−5 +i 1≤i≤n−1.

Obviously the above vertex labeling is a difference cordial labeling ofS(DTn).

A double quadrilateral snakeDQnconsists of two

tri-angular snakes that have a common path.

Theorem 3.4 S(DQn)is difference cordial.

Proof. Let V(S(DQn)) = V(DQn) ∪

{

u′_i, v_i′, w′_i, z_i′, x′_i, y_i′, zi: 1≤i≤n−1

}

and E(S(DQn)) =

{

uiu

′

i, u

′

iui+1: 1≤i≤n−1

} ∪ {

uiv

′

i, uix

′

i, v

′

ivi, x

′

ixi, viz

′

i, xizi, z

′

iwi: 1≤i≤n−1

} ∪ {

ziyi, wiw

′

i, yiy

′

i, w

′

iui+1, y

′

iui+1: 1≤i≤n−1

}

. De-fine a map f : V(S(DQn)) → {1,2, . . . ,12n−11}

by

f(ui) = 6i−5 1≤i≤n

f

(

v_i′

)

= 6i−4 1≤i≤n−1 f(vi) = 6i−3 1≤i≤n−1

f

(

zi′

)

= 6i−2 1≤i≤n−1 f(wi) = 6i−1 1≤i≤n−1

f

(

w′_i

)

= 6i 1≤i≤n−1

f

(

x′_i

)

= 6n+ 2i−6 1≤i≤n−1 f(xi) = 6n+ 2i−5 1≤i≤n−1

f(zi) = 8n+i−7 1≤i≤n−1

f(yi) = 9n+i−8 1≤i≤n−1

f

(

y_i′

)

= 10n+i−9 1≤i≤n−1 f

(

u′_i

)

= 11n+i−10 1≤i≤n−1. Since ef(0) =ef(1) = 7n−7, f is a difference cordial

labeling of S(DQn).

An alternate triangular snakeA(Tn) is obtained from

a pathu1u2. . . , unby joininguiandui+1(alternatively)

to new vertexvi. That is every alternate edge of a path

is replaced byC3.

Theorem 3.5 S(A(Tn))is difference cordial.

Proof. Let the edgesuiui+1, uivi andviui+1 be

subdi-vided byu′_i, xi andyirespectively.

Case 1. Let the triangle be starts fromu1 and ends

withun.

In this case S(A(Tn)) consists of 7n2−2 vertices and

4n− 2 edges. Define a map f : V(S(A(Tn))) →

{

1,2, . . . ,7n₂−2}by

f(ui) = 2i−1 1≤i≤n

f

(

u′_i

)

= 2i 1≤i≤n−1

f(xi) = 2n−1 +i 1≤i≤

n 2 f(yi) =

5n−2

2 +i 1≤i≤

n 2 f(vn

2−i+1 )

= 3n−1 +i 1≤i≤n 2. Since ef(0) =ef(1) = 2n−1, f is a difference cordial

labeling of S(A(Tn)).

Case 2. Let the triangle be starts fromu2 and ends

withun−1.

In this case, S(A(Tn)) has 7n₂−8 vertices and 4n−6

edges. Define an injective map f : V (S(A(Tn))) →

{

1,2, . . . ,7n−8 2

}

by

f(ui) = 2i−2 2≤i≤n

f

(

u′_i

)

= 2i−1 1≤i≤n−1 f(xi) = 2n−2 +i 1≤i≤

n−2 2 f(yi) =

5n−6

2 +i 1≤i≤

n−2 2 f(vi) = 3n−3 +i 1≤i≤

f(u1) = 3n−3. Obviously, f is a difference cordial

labeling ofS(A(Tn)).

Case 3. Let the triangle be starts from u2 and ends

withun.

In this case, the order and size of S(A(Tn)) are 7n₂−5

and 4n−4 respectively. The difference cordial labeling ofS(A(T3)) is given in figure 1.

8 1 2 3 4

5 7

6

Figure 1

Forn >3, Define an injective mapf :V(S(A(Tn)))→

{

1,2, . . . ,7n−5 2

}

by

f(ui) = 2i−1 1≤i≤n

f

(

u′i

)

= 2i 1≤i≤n−1

f(xi) = 2n−1 +i 1≤i≤

n−1 2 f(yi) =

5n−3

2 +i 1≤i≤

n−1 2 f(vi) = 3n−2 +i 1≤i≤

n−1 2 . Sinceef(0) =ef(1) = 2n−2,f is a difference cordial

labeling ofS(A(Tn)).

A double alternate triangular snake DA(Tn)

con-sist of two triangular snakes that have a common path. That is, a double alternate triangular snake is obtained from a pathu1u2. . . un by joiningui andui+1

(alternatively) to two new verticesvi andwi.

Theorem 3.6 S(DA(Tn))is difference cordial.

Proof. Let the edgesuiui+1,uivi,viui+1,uiwi,wiui+1

be subdivided byu′_i,xi,yi,x

′

i,y

′

i respectively.

Case 1. Let the two triangles be starts from u1 and

ends withun.

Here, the number of vertices and edges inS(DA(Tn))

are 5n−1 and 6n−2 respectively. Define a map f : V(S(DA(Tn)))→ {1,2, . . . ,5n−1}by

f(ui) = 2i−1 1≤i≤n

f

(

u′_i

)

= 2i 1≤i≤n−1

f(xi) = 2n−2 + 2i 1≤i≤

n 2 f(vi) = 2n−1 + 2i 1≤i≤

n 2 f(yn

2−i+1 )

= 3n−1 +i 1≤i≤ n 2 f

(

x′_i

)

= 7n−2

2 +i 1≤i≤

n 2 f

(

y_i′

)

= 4n−1 +i 1≤i≤ n 2 f(wi) =

9n−2

2 +i 1≤i≤

n 2. Sinceef(0) =ef(1) = 3n−1,f is a difference cordial

labeling ofS(DA(Tn)).

Case 2. Let the two triangles be starts fromu2 and

ends withun−1.

In this case, the order and size of S(DA(Tn)) are

5n−7 and 6n−10 respectively. Define a function f : V (S(DA(Tn)))→ {1,2, . . . ,5n−7}byf(u1) = 2n−1,

f(ui) = 2i−2 2≤i≤n

f

(

u′_i

)

= 2i−1 1≤i≤n−1 f(xi) = 2n−2 + 2i 1≤i≤

n−2 2 f(vi) = 2n−1 + 2i 1≤i≤

n−2 2 f(yi) = 4n−5 +i 1≤i≤

n−2 2 f

(

x′_i

)

= 3n+ 2i−4 1≤i≤n−2 2 f

(

y′_i

)

= 9n−12

2 +i 1≤i≤

n−2 2 f(wi) = 3n+ 2i−3 1≤i≤

n−2 2 .

Since ef(0) =ef(1) = 3n−5,f is a difference cordial

labeling ofS(DA(Tn)).

Case 3. Let the two triangles be starts from u2 and

ends withun.

In this case, the order and size of S(DA(Tn)) consists

of 5n−4 vertices and 6n−6 edges respectively. Define a functionf :V(S(DA(Tn)))→ {1,2, . . . ,5n−4}by

f(ui) = 2i−1 1≤i≤n

f

(

u′i

)

= 2i 1≤i≤n−1

f(xi) = 2n+ 2i−2 1≤i≤

n−1 2 f(vi) = 2n+ 2i−1 1≤i≤

n−1 2 f

(

x′_i

)

= 3n+ 2i−3 1≤i≤n−1 2 f(wi) = 3n+ 2i−2 1≤i≤

n−1 2 f(yi) = 4n−3 +i 1≤i≤

n−1 2 f

(

y_i′

)

= 9n−7

2 +i 1≤i≤

n−1 2 .

Since ef(0) =ef(1) = 3n−3,f is a difference cordial

labeling ofS(DA(Tn)).

An alternate quadrilateral snake A(Qn) is obtained

from a path u1u2. . . un by joining ui, ui+1

(alterna-tively) to new vertices vi, wi respectively and then

joining vi and wi. That is every alternate edge of a

path is replaced by a cycleC4.

Theorem 3.7 S(AQn)is difference cordial.

Proof. Let the edgesuiui+1,uivi,viwi,wiui+1be

sub-divided byu′_i,v′_i,zi,w

′

i respectively.

Case 1. Let the squares be starts from u1 and ends

withun.

In this case the order and size ofS(AQn) are 9n₂−2 and

{

1,2, . . . ,9n₂−2}as follows:

f(ui) = 2i−1 1≤i≤n

f

(

u′i

)

= 2i 1≤i≤n−1

f

(

v_i′

)

= 2n+ 2i−2 1≤i≤ n 2 f(vi) = 2n+ 2i−1 1≤i≤

n 2 f

(

wi′

)

= 3n−1 +i 1≤i≤ n 2 f(zi) =

7n−2

2 +i 1≤i≤

n 2 f(wn

2−i+1 )

= 4n−1 +i 1≤i≤ n 2. Since ef(0) = ef(1) = 5n₂−2, f is a difference cordial

labeling ofS(AQn).

Case 2. Let the squares be starts from u2 and ends

withun−1.

The difference cordial labeling of S(AQ4) is given in

figure 2.

10 12 11

7 1 2 3 4 5 6

[image:4.595.62.558.49.572.2]

8 9

Figure 2

For n > 4, Define a map f : V(S(AQn)) →

{

1,2, . . . ,9n−₂12}by

f(ui) = 2i−2 2≤i≤n

f

(

u′_i

)

= 2i−1 1≤i≤n−1 f

(

v_i′

)

= 2n+ 2i−3 1≤i≤ n−2 2 f(vi) = 2n+ 2i−2 1≤i≤

n−2 2 f

(

wi′

)

= 3n−4 +i 1≤i≤ n−2 2 f(zi) =

7n−10

2 +i 1≤i≤

n−2 2 f(wi) = 4n−6 +i 1≤i≤

n−2 2

andf(u1) =9n−₂12. Sinceef(0) =ef(1) = 5n−8,f is

a difference cordial labeling ofS(AQn).

Case 3. Let the squares be starts from u2 and ends

withun.

The difference cordial labeling of S(AQ3) is given in

figure 3.

8 9 7

1 2 3 4 5

6 10

Figure 3

For n > 3, Define a map f : V (S(AQn)) →

{

1,2, . . . ,9n₂−7}by

f(ui) = 2i−1 1≤i≤n

f

(

u′i

)

= 2i 1≤i≤n−1

f

(

v′_i

)

= 2n+ 2i−2 1≤i≤n−1 2 f(vi) = 2n+ 2i−1 1≤i≤

n−1 2 f(zi) = 3n−2 +i 1≤i≤

n−1 2 f(wi) =

7n−5

2 +i 1≤i≤

n−1 2 f

(

w′_i

)

= 4n−3 +i 1≤i≤n−1 2 . Since ef(0) = ef(1) = 5n₂−5, f is a difference cordial

labeling of S(AQn).

Illustration 1 The difference cordial labeling of

DA(Q8)is given in figure 4.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

16 24 18 25 20 26 22 27

17 28 35 19 29 34 21 30 33 23 31 32

Figure 4

The difference cordial labeling of DA(Q8) is given in

figure 5.

30 1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 21 17 22 19 23

16 24 27 18 25 28 20 26 29

Figure 5

The difference cordial labeling of DA(Q7) is given in

figure 6.

1 2 3 4 5 6 7 8 9 10 11 12 13

14 26 16 27 18 28

15 20 23 17 21 24 19 22 25

Figure 6

A double alternate quadrilateral snake DA(Qn)

con-sists of two alternate quadrilateral snakes that have a common path. That is, a double alternate quadrilateral snake is obtained from a path u1u2. . . un by joining ui

and ui+1 (alternatively) to new vertices vi, xi and wi,

yi respectively and then joiningvi,wi andxi,yi.

Theorem 3.8 S(DAQn)is difference cordial.

Proof. Let the egdes uiui+1, uivi, viwi, wiui+1, uixi,

xiyi, yiui+1 be subdivided by u

′

i, v

′

i, zi, w

′

i, x

′

i, z

′

i, y

′

i

respectively.

Case 1. Let the squares be starts from u1 and ends

withun.

respectively. Define a map f : V(S(DAQn)) →

{1,2, . . . ,7n−1} by

f(ui) = 2i−1 1≤i≤n

f

(

u′_i

)

= 2i 1≤i≤n−1

f(vi) = 2n+ 3i−3 1≤i≤

n 2 f(zi) = 2n+ 3i−2 1≤i≤

n 2 f(wi) = 2n+ 3i−1 1≤i≤

n 2 f(xi) =

7n−6

2 + 3i 1≤i≤ n 2 f

(

z_i′

)

= 7n−4

2 + 3i 1≤i≤ n 2 f(yi) =

7n−2

2 + 3i 1≤i≤ n 2 f

(

x′_i

)

= 11n−2

2 +i 1≤i≤ n 2 f

(

w_i′

)

= 13n−2

2 +i 1≤i≤ n 2 f

(

y′n

2−i+1 )

= 5n−1 +i 1≤i≤ n 2 f

(

vi′

)

= 6n−1 +i 1≤i≤ n 2. Sinceef(0) =ef(1) = 4n−1,f is a difference cordial

labeling ofS(DAQn).

Case 2. Let the squares be starts from u2 and ends

withun₋1.

The order and size of S(DAQn) are 7n − 11 and

8n−14 respectively. Define a mapf :V (S(DAQn))→

{1,2, . . . ,7n−11}by

f(ui) = 2i−2 2≤i≤n

f

(

u′_i

)

= 2i−1 1≤i≤n−1

f(vi) = 2n+ 3i−4 1≤i≤

n−2 2 f(zi) = 2n+ 3i−3 1≤i≤

n−2 2 f(wi) = 2n+ 3i−2 1≤i≤

n−2 2 f(xi) =

7n−14

2 + 3i 1≤i≤

n−2 2 f

(

z_i′

)

= 7n−12

2 + 3i 1≤i≤

n−2 2 f(yi) =

7n−10

2 + 3i 1≤i≤

n−2 2 f

(

w_i′

)

= 11n−18

2 +i 1≤i≤

n−2 2 f

(

y_i′

)

= 13n−22

2 +i 1≤i≤

n−2 2 f

(

v_i′

)

= 5n−8 +i 1≤i≤n−2 2 f

(

x′_i

)

= 6n−10 +i 1≤i≤n−2 2 . andf(u1) = 7n−11. Sinceef(0) =ef(1) = 4n−7,f

is a difference cordial labeling ofS(DAQn).

Case 3. Let the squares be starts from u2 and ends

withun.

Assign the labels to the vertices ui (1≤i≤n),

u′_i (1≤i≤n−1),vi,zi,wi

(

1≤i≤ n−1 2

)

as in case 1

and define f(xi) =

7n−9

2 + 3i 1≤i≤ n−1

2 f

(

z_i′

)

= 7n−7

2 + 3i 1≤i≤ n−1

2 f(yi) =

7n−5

2 + 3i 1≤i≤ n−1

2 f

(

w′_i

)

= 11n−9

2 +i 1≤i≤ n−1

2 f

(

y′_i

)

= 13n−11

2 +i 1≤i≤ n−1

2 f

(

v′_i

)

= 5n−4 +i 1≤i≤ n−1 2 f

(

x′i

)

= 6n−5 +i 1≤i≤ n−1 2 . Since ef(0) =ef(1) = 4n−4,f is a difference cordial

labeling ofS(DAQn).

4

Conclusion

In this paper we have proved that subdivision of quadrilateral snakes, triangular snakes, Alternate triangular snakes, double triangular snakes, Alternate quadrilateral snakes, Double quadrilateral snakes, Double alternate triangular snakes, double alternate quadrilateral snakes are difference cordial. Investigation of difference cordial labeling behavior of subdivision of G1∪G2, G1+G2, G1×G2 etc are the open problems

for future research.

Acknowledgement

The authors thank the refree for his valuable sugges-tions and comments.

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