On Z 2 x Z 2 - Cordial Graphs

On

2

2







- Cordial Graphs

S. Beena

Associate Professor, Department of Mathematics, NSS College, Nilamel, Kollam, Kerala, India

Abstract

For any abelian group

A

, a graph G is said to be A - cordial if there is a labeling f of V(G) with elements of A so that for all a,bA _{, the edge ab is labeled with} f(a)  f(b)then the number of vertices labeled with a and the vertices labeled with b differ by at most 1and the number of edges labeled with a and edges labeled with b differ by at most 1. In this paper we determine some classes of 22 cordial graphs, a

necessary condition for the sum of two 22- cordial graphs to be 22- cordial and we prove that every

graph is an induced sub graph of 22- cordial graph.

Key words - A- Cordial labeling, Abelian group.

I. INTRODUCTION

Mark Hovey [1] has introduced A – cordial labeling as a generalization of harmonious and cordial labeling. It is well known that 2

 

0,1 is the residue classes modulo 2, where 0 denote the set of all even integers and 1 denote the set of all odd integers and consequently 22



       

0,0,1,0, 0,1,1,1



is an abelian

group. Without loss of generality, we may assume thate

 

0,0,

a



 

1

,

0

,

b



 

0

,

1

,

c



 

1

,

1

. Then

b

c

a

c

b

a





,





and

b



c



a

. In this paper we determine some classes of 22cordial graphs and a necessary condition for sum of two 22- cordial graphs to be 22- cordial. Also we prove that every graph is an induced subgraph of 22- cordial graph.

Definition 1. For any abelian group

A

, a graph

G





V

,

E



is said to be

A

- cordial , if there is a labeling

f

of

V

with elements of

A

so that for all

a

,

b



A

, the edge

ab

is labeled with

f

(

a

)



f

(

b

)

then

v

f

(

a

)

and

)

(

b

v

f differ by at most 1 and

e

f

(

a

)

and

e

f

(

b

)

differ by at most 1, where

v

f

(

a

)

and

e

f

(

b

)

are respectively the number of vertices labeled with

a

and the number of edges labeled with

b

.

II. MAIN RESULTS

Theorem 2: The cycle C is n 22- cordial for all n except n4,5and n2(mod4).

Proof: Let V

 

C_n 



v_i:1in



be the set of all vertices of

C

n, and E

 

Cn 



ei vivi1:1in1



be the

set of all edges

of C . Define n f :V

 

Cn 22 as follows: Case(i) If n0,1,3,7(mod8): Label the vertices of Cn as

e v

f( i) , if i4k,4k1 where

k

is odd, f(vi)a, if i1,8k,8k1 where

k

is an integer, f(vi)b, if )

4 (mod 2



i and f(vi)c, if i3(mod4).

Case(ii) If n4 or 5(mod8)where n4,5: Label the vertices of Cn as f(vi)e if i4,5,8,8k,8k1 where k1is an integer, f(vi)a, if i1,7,4k,4k1 where k1is odd, f(vi)b if i2,9,2k where

3



k is odd, f(vi)c, if i3,6,4k3 where k1is an integer. Table 1 shows the values of v_f(x)for all

2 2

 

x and Table 2 shows the values of ef(x)for all x22. Alsovf(i)vf(j) 1and 1

) ( )

(i e j 

ef f for all i,j22. HenceCnis 22- cordial for all

n

except n4,5and )

4 (mod 2



Table 1. Labelings of the vertices of C_n

Table 2. Labelings of the edges of C_n

Theorem3. The complete bipartite graph Km,nwhere mn is 22- cordial for all

m

and

n

except )

4 (mod 2 &n 

m .

Proof: Let V1



ui:1im



and V2 



vj :1 jn



be the set vertices of Km,n and



e uv i m j n



K

E( _m,n) _i  _{i j}:1  ,1  be the set of all edges of Km,n. Define f :V(Km,n)22 as follows: f(ui)e if i1(mod4), f(ui)a if i2(mod4), f(ui)b if i  3(mod4)

,

f(ui)c if

) 4 (mod 0



i .

Case(i) If m0,1,2(mod4) 𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 𝑖𝑓 m2(mod4), 𝑐ℎ𝑜𝑜𝑠𝑒

n

𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡

n

≢ 2(𝑚𝑜𝑑4) , define

e v

f( i) if ) 4 (mod 0



i , f(v_i)a if i3(mod4),f(vi)b if i1(mod4),f(v_i)c if i2(mod4).

If m4r,Table 3 and 4 shows the values of vf(x) and ef(x)for all x22.

Table 3. Labelings of the vertices of K_m,n

.

Table 4. Labelings of the edges of K_m,n

If m4r1Table 5 and 6 shows the values of v_f(x) and e_f(x)for all x₂₂.

i r

n8  v_f(e) v_f(a) v_f(b) v_f(c)

0



i 2𝑟 2𝑟 2𝑟 2𝑟

1



i 2𝑟 2𝑟 + 1 2𝑟 2𝑟

3



i 2𝑟 2𝑟 + 1 2𝑟 + 1 2𝑟 + 1

4



i 2𝑟 + 1 2𝑟 + 1 2𝑟 + 1 2𝑟 + 1

5



i 2𝑟 + 1 2𝑟 + 2 2𝑟 + 1 2𝑟 + 1

7



i 2𝑟 + 2 2𝑟 + 1 2𝑟 + 2 2𝑟 + 2

i r

n8  e_f(e) e_f(a) e_f(b) e_f(c)

0



i 2𝑟 2𝑟 2𝑟 2𝑟

1



i 2r+1 2𝑟 2𝑟 2𝑟

3



i 2𝑟 2𝑟 + 1 2𝑟 + 1 2𝑟 + 1

4



i 2𝑟 + 1 2𝑟 + 1 2𝑟 + 1 2𝑟 + 1

5



i 2𝑟 + 2 2𝑟 + 1 2𝑟 + 1 2𝑟 + 1

7



i 2𝑟 + 1 2𝑟 + 2 2𝑟 + 2 2𝑟 + 2

i k

n4  vf(e) vf(a) vf(b) vf(c)

0



i 𝑟 + 𝑘 + 1 𝑟 + 𝑘 𝑟 + 𝑘 𝑟 + 𝑘

1



i 𝑟 + 𝑘 𝑟 + 𝑘 𝑟 + 𝑘 + 1 𝑟 + 𝑘

2



i 𝑟 + 𝑘 𝑟 + 𝑘 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1

3



i 𝑟 + 𝑘 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1

i k

n4  e_f(e) e_f(a) e_f(b) e_f(c)

0



i nr nr nr nr

1



i nr nr nr nr

2



i nr nr nr nr

3



i nr nr nr nr

i k

n4  v_f(e) v_f(a) v_f(b) v_f(c)

0



i 𝑟 + 𝑘 + 1 𝑟 + 𝑘 𝑟 + 𝑘 𝑟 + 𝑘

1



Table5. Labelings of the vertices of K_m,n

Table 6. Labelings of the edges of K_m,n

If m4r2Table 7 and 8 shows the values of v_f(x) and e_f(x)for all x22.

Table7. Labelings of the vertices of K_m,n

Table8. Labelings of the edges of K_m,_n

If m4r3Table 9 and 10 shows the values of v_f(x) and e_f(x)for all x22.

Table 9. Labelings of the vertices of K_m,n

Table 10. Labelings of the edges of K_m,n

Case(ii) If m3(mod4), define f(vi)e if i2(mod4), f(vi)a if i3(mod4),f(vi)b if )

4 (mod 0



i ,

c v

f( i) if i1(mod4).

If m4r3Table 11 and 12 shows the values of vf(x) and ef(x)for all x22 2



i 𝑟 + 𝑘 + 1 𝑟 + 𝑘 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1

3



i 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1

i k

n4  e_f(e) e_f(a) e_f(b) e_f(c)

0



i 𝑚𝑘 𝑚𝑘 𝑚𝑘 𝑚𝑘

1



i 𝑚𝑘 + 𝑟 𝑚𝑘 + 𝑟 𝑚𝑘 + 𝑟 + 1 𝑚𝑘 + 𝑟

2



i 𝑚𝑘 + 2𝑟 𝑚𝑘 + 2𝑟 𝑚𝑘 + 2𝑟 + 1 𝑚𝑘 + 2𝑟 + 1

3



i 𝑚𝑘 + 3𝑟 𝑚𝑘 + 3𝑟 + 1 𝑚𝑘 + 3𝑟 + 1 𝑚𝑘 + 3𝑟 + 1

i k

n4  v_f(e) v_f(a) v_f(b) v_f(c)

0



i 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 𝑟 + 𝑘

1



i 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘

3



i 𝑟 + 𝑘 𝑟 + 𝑘 + 1 𝑟 + 𝑘 𝑟 + 𝑘 + 1

i k

n4  e_f(e) e_f(a) e_f(b) e_f(c)

0



i 𝑚𝑘 𝑚𝑘 𝑚𝑘 𝑚𝑘

1



i 𝑚𝑘 + 𝑟 𝑚𝑘 + 𝑟 𝑚𝑘 + 𝑟 + 1 𝑚𝑘 + 𝑟 + 1

3



i 𝑚𝑘 + 3𝑟 + 1 𝑚𝑘 + 3𝑟 + 1 𝑚𝑘 + 3𝑟 + 2 𝑚𝑘 + 3𝑟 + 2

i k

n4  v_f(e) v_f(a) v_f(b) v_f(c)

0



i 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 1



i 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1

2



i 𝑟 + 𝑘 + 2 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1

3



i 𝑟 + 𝑘 + 2 𝑟 + 𝑘 + 2 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1

i k

n4  ef(e) ef(a) ef(b) ef(c)

0



i 𝑚𝑘 𝑚𝑘 𝑚𝑘 𝑚𝑘

1



i 𝑚𝑘 + 𝑟 𝑚𝑘 + 𝑟 + 1 𝑚𝑘 + 𝑟 + 1 𝑚𝑘 + 𝑟 + 1

2



i 𝑚𝑘 + 2𝑟 + 1 𝑚𝑘 + 2𝑟 + 2 𝑚𝑘 + 2𝑟 + 2 𝑚𝑘 + 2𝑟 + 1

3



i 𝑚𝑘 + 3𝑟 + 2 𝑚𝑘 + 3𝑟 + 3 𝑚𝑘 + 3𝑟 + 2 𝑚𝑘 + 3𝑟 + 2

i k

n4  v_f(e) v_f(a) v_f(b) v_f(c)

0



i 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘

1



Table 11. Labelings of the vertices of K_m,n

Table 12. Labelings of the edges of K_m,n

In all cases, vf(i)vf(j) 1and ef(i)ef(j) 1 for all i,j22. Hence the complete bipartite graph

n m

K _, where

m



n

is 22- cordial for all m and nexceptm&n2(mod4).

Remark: 22- cordiality of

K

m,n IS found in [2] . However, the proof is different from ours.

Theorem 4. Let Giare (pi,qi)22- cordial labeled graph under fi for i1,2respectively. Then G1G2 is 22- cordial if (i) either p1 or p2 0(mod4)and (ii) either q1 or q2 0(mod4).

Proof: Case (i) Let p10(mod4)and q10(mod4). Then

4 ) ( 1 1 p i

vf  ,

4 ) ( 1 1 q i

ef  ie,a,b,c. Also,

1 ) ( )

( ₂

2 i v j 

vf f and ef2(i)ef2(j) 1 i,je,a,b,c. Let vf2(e)m1, vf2(a)m2, vf2(b)m3,

4

) (

2 c m

v_f  ,

(

)

₁

2

e

n

e

f



, ef2(a)n2, ef2(b)n3, and vf2(c)m4. Define f :V(G1G2)22

such that f |V(Gi) fi fori1,2. Then clearly vf(i)vf(j) 1 i,je,a,b,cand

, 4 4

)

( 1 1 2

1 p p

n q e

ef    ,

4 4

)

( 2 1 2

1 p p

n q a

ef    ,

4 4

)

( 3 1 2

1 p p

n q b

ef   

4 4

)

( 4 1 2

1 p p

n q c

ef   

.Therefore, ef(i)ef(j) ef₂(i)ef₂(j) 1 i,je,a,b,c.

Case(ii). Let p10(mod4)and q20(mod4). Then

4 ) ( 1 1 p i

vf  , ie,a,b,candef₁(i)ef₁(j) 1,

c b a e j

i,  , , ,

 . Let ef1(e)n1, ef1(a)n2, ef1(b)n3, ef1(c)n4and vf2(i)vf2(j) 1 ,

c b a e j

i,  , , ,

 and ef i q i e,a,b,c

4 )

( 2

2    . Let

v

f2

(

e

)



m

1,

v

f2

(

a

)



m

2,

v

f2

(

b

)



m

3and

4

) (

2 c m

v_f  . Define f :V(G1G2)22 such that f |V(Gi) fi fori1,2.Then 1 1

4 )

(e p m

v_f   ,

2 1

4 )

(a p m

v_f   , 3

1

4 )

(b p m

v_f   and 4

1

4 )

(c p m

v_f   .Therefore, vf(i)vf(j) vf₂(i)vf₂(j) 1

c

b

a

e

j

i

,



,

,

,



. Now,

4 4 ) ( 4 4 )

( 1 2 3 4 1 1 1 2

1 1 1 p p q n m m m m p q n e

ef          . Similarly

4 4 )

( 2 1 1 2

p p q n a

ef    ,

4 4 )

( 3 1 1 2

p p q n b

ef    and

4 4 )

( 4 1 1 2

p p q n c

ef    .Therefore,

1 ) ( ) ( ) ( ) ( 1

1  



e j e i e j i

ef f f f



i

,

j



e

,

a

,

b

,

c

. In a similar manner we can consider the other two cases. Hence G1G2 is 22- cordial under f .

Theorem5. Every connected graph is an induced sub graph of a 22- cordial graph.

Proof: Let Gbe a connected

(

p

,

q

)

graph with vertices

v

₁

,

v

₂

,...,

v

_p. If G _{itself is} 22- cordial, there is nothing to prove. Otherwise, define f :V(G)22such that vf(i)vf(j) 1and ef(i)ef(j) 

c

b

a

e

j

i

,



,

,

,



where



is possibly a small positive integer. Let ef(e)m1, ef(a)m2, ef(b)m3, 2



i 𝑟 + 𝑘 + 2 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1

3



i 𝑟 + 𝑘 + 2 𝑟 + 𝑘 + 2 𝑟 + 𝑘 + 1 𝑟 + 𝑘 + 1

i k

n4  ef(e) ef(a) ef(b) ef(c)

0



i 𝑚𝑘 𝑚𝑘 𝑚𝑘 𝑚𝑘

1



i 𝑚𝑘 + 𝑟 𝑚𝑘 + 𝑟 + 1 𝑚𝑘 + 𝑟 + 1 𝑚𝑘 + 𝑟 + 1

2



i 𝑚𝑘 + 2𝑟 + 1 𝑚𝑘 + 2𝑟 + 2 𝑚𝑘 + 2𝑟 + 2 𝑚𝑘 + 2𝑟 + 1

3



4

) (c m

e_f  . Let mmin{m1,m2,m3,m4} and M max{m1,m2,m3,m4}. Since G is not 22 cordial, we have Mm2 and p3. Let

u

be a vertex in G whose label is

x

, and such that

v

_f

(

x

)



n

₁ is the minimum. Add a new vertex

v

and label it withx, then

v

_f

(

x

)



n

₁



1

. Let Mm2. If Mm11, join

v

with a vertex in G whose label is

x

, then

e

_f

(

e

)



m

₁



1

. If Mm2 1, join

v

with a vertex in Gwhose label is xa, thene_f(a)m₂ 1 . Similarly if Mm31, join

v

with a vertex in Gwhose label is

x



b

, then

e

f

(

b

)



m

3



1

and if if Mm4 1, join

v

with a vertex in Gwhose label is

x



c

, then

1

)

(

c



m

₄



e

_f . If Mm2, we repeat the above process with the new graph H1. Since this process reduce the difference Mm, after a finite number of steps, we get a 22- cordial graph which contains G as an induced sub graph.

Notation : A (p,q)graph has

p

vertices and

q

edges. For basic concepts in graph theory, we follow [3] and for graph labeling follow [4].

CONCLUSION

The cycle Cnis 22- cordial for all nexcept n4,5and n2(mod4)and the complete bipartite graph Km,nwhere mn is 22- cordial for all m and nexceptm& n  2(mod4). Also if Giare

) ,

(pi qi 22- cordial labeled graph under fi for i1,2respectively then G1G2 is 22- cordial if (i) either p1 or p2 0(mod4)and (ii) either q1 or q2 0(mod4). Every connected graph is an induced sub graph of a 22- cordial graph.

ACKNOWLEDGEMENTS

The author is grateful to Dr M I Jinnah, Former Prof & Head, Department of Mathematics, University of Kerala, Thiruvananthapuram for his valuable suggestions.

REFERENCES

[1] Mark Hovey, A – cordial Graphs, Discrete mathematics 93(1991) 183 – 194, North Holland.

[2] O.Pechenik and J. Wise, Generalized graph cordialty, Discuss. Math. Graph Theory 32 (2012) 557-567. [3] WB West, Introduction to Graph Theory, (2nd _{Edn) Pearson Education (2001)}