Some New Families of 4-Prime Cordial Graphs

2017·0.16 MB·

by  R. Ponraj

International J.Math. Combin. Vol.3(2017), 125-135
Some New Families of 4-Prime Cordial Graphs
R.Ponraj
(DepartmentofMathematics,SriParamakalyaniCollege,Alwarkurichi-627412,India)
Rajpal Singh and R.Kala
(DepartmentofMathematics,ManonmaniamSundaranarUniversity,Tirunelveli-627012,India)
E-mail: [email protected],[email protected],[email protected]
Abstract: Let G be a (p,q) graph. Let f : V(G) 1,2,...,k be a function. For
→ { }
each edge uv, assign the label gcd(f(u),f(v)). f is called k-prime cordial labeling of G
if v (i) v (j) 1, i,j 1,2,...,k and e (0) e (1) 1 where v (x) denotes the
f f f f f
| − | ≤ ∈ { } | − | ≤
numberof vertices labeled with x, e (1) and e (0) respectively denote thenumberof edges
f f
labeled with 1 and not labeled with 1. A graph with admits a k-prime cordial labeling is
calledak-primecordialgraph. Inthispaperweinvestigate4-primecordiallabelingbehavior
of shadow graph of a path, cycle, star, degree splitting graph of a bistar, jelly fish,splitting
graph of a path and star.
Key Words: Cordiallabeling,Smarandachelycordiallabeling,cycle,star,bistar,splitting
graph.
AMS(2010): 05C78.
§1. Introduction
In this paper graphs are finite, simple and undirected. Let G be a (p,q) graph where p is the
numberofverticesofGandqisthenumberofedgeofG. In1987,Cahitintroducedtheconcept
of cordial labeling of graphs [1]. Sundaram, Ponraj, Somasundaram [5] have been introduced
thenotionofprimecordiallabelinganddiscussedtheprimecordiallabelingbehaviorofvarious
graphs. Recently Ponraj et al. [7], introduced k-prime cordial labeling of graphs. A 2-prime
cordial labeling is a product cordial labeling [6]. In [8, 9] Ponraj et al. studied the 4-prime
cordiallabelingbehaviorofcompletegraph,book,flower,mC ,wheel,gear,doublecone,helm,
n
closed helm, butterfly graph, and friendship graph and some more graphs. Ponraj and Rajpal
singhhavestudiedaboutthe4-primecordialityofunionoftwobipartitegraphs,unionoftrees,
durergraph,tietze graph,planargridP P , subdivisionofwheelsandsubdivisionofhelms,
m n
×
lotus inside a circle,sunflowergraphandthey have obtainedsome 4-prime cordialgraphs from
4-prime cordial graphs [10, 11, 12]. Let x be any real number. In this paper we have studied
about the 4-prime cordiality of shadow graph of a path, cycle, star, degree splitting graph of a
1ReceivedNovember12,2016,Accepted August28,2017.
126 R.Ponraj,RajpalSinghandR.Kala
bistar,jellyfish,splitting graphofapathandstar. Letxbeanyrealnumber. Then x stands
⌊ ⌋
for the largest integer less than or equal to x and x stands for smallest integer greater than
⌈ ⌉
or equal to x. Terms not defined here follow from Harary [3] and Gallian [2].
§2. k-Prime Cordial Labeling
Let G be a (p,q) graph. Let f : V(G) 1,2, ,k be a map. For each edge uv, assign
→ { ··· }
the label gcd(f(u),f(v)). f is called k-prime cordial labeling of G if v (i) v (j) 1, i,j
f f
| − |≤ ∈
1,2, ,k and e (0) e (1) 1, and conversely, if v (i) v (j) 1, i,j 1,2, ,k
f f f f
{ ··· } | − | ≤ | − | ≥ ∈ { ··· }
or e (0) e (1) 1, it is called a Smarandachely cordial labeling, where v (x) denotes the
f f f
| − | ≥
number of vertices labeled with x, e (1) and e (0) respectively denote the number of edges
f f
labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is called a
k-prime cordial graph.
First we investigate the 4-prime cordiality of shadow graph of a path, cycle and star. A
shadow graphD (G) of a connected graph G is constructed by taking two copies of G, G and
2 ′
G and joining each vertex u in G to the neighbors of the corresponding vertex u in G .
′′ ′ ′ ′′ ′′
Theorem 2.1 D (P ) is 4-prime cordial if and only if n=2.
2 n
6
Proof It is easy to see that D (P ) is not 4-prime cordial. Consider n > 2. Let
2 2
V(D (P )) = u ,v : 1 i n and E(D (P )) = u u ,v v : 1 i n 1
2 n i i 2 n i i+1 i i+1
{ ≤ ≤ } { ≤ ≤ − } ∪
u v ,v u : 1 i n 1 . In a shadow graph of a path, D (P ), there are 2n vertices
i i+1 i i+1 2 n
{ ≤ ≤ − }
and 4n 4 edges.
−
Case 1. n 0 (mod 4).
≡
Let n = 4t. Assign the label 4 to the vertices u ,u , ,u then assign 2 to the
1 2 2t
···
vertices v ,v , ,v . For the vertices v ,v , we assign 3,1 respectively. Put the
1 2 2t 2t+1 2t+2
···
label 1 to the vertices v ,v , ,v . Now we assign the label 3 to the vertices
2t+3 2t+5 4t 1
··· −
v ,v , ,v . Then assignthe label1 to the vertexv . Nextwe consider the vertices
2t+4 2t+6 4t 2 4t
··· −
u ,u , ,u . Put3,3totheverticesu ,u . Thenfixthenumber1tothevertices
2t+1 2t+2 4t 2t+1 2t+2
···
u ,u , ,u . Finally assign the label 3 to the vertices u ,u , ,u .
2t+3 2t+5 4t 1 2t+4 2t+6 4t
··· − ···
Case 2. n 1 (mod 4).
≡
Taken=4t+1. Assignthelabel4totheverticesu ,u , ,u . Thenassignthelabel3
1 2 2t+1
···
totheverticesu ,u , ,u andputthenumber1totheverticesu ,u , ,u .
2t+2 2t+4 4t 2t+3 2t+5 4t+1
··· ···
Nextweturntotheverticesv ,v , ,v . Assignthelabel2totheverticesv ,v , ,v .
1 2 2t+1 1 2 2t+1
··· ···
The remaining vertices v (2t+2 i 4t+1) are labeled as in u (2t+2 i 4t+1).
i i
≤ ≤ ≤ ≤
Case 3. n 2 (mod 4).
≡
Letn=4t+2. Assignthe labels tothe verticesu ,v (1 i 2t+1)asincase2. Nowwe
i i
≤ ≤
consider the vertices u ,u , ,u . Assign the labels 3,1 to the vertices u ,u
2t+2 2t+3 4t+2 2t+2 2t+3
···
respectively. Then assign the label 1 to the vertices u ,u , ,u . Put the integer
2t+4 2t+6 4t+2
···
3 to the vertices u ,u , ,u . Now we turn to the vertices v ,v , ,v .
2t+5 2t+7 4t+1 2t+2 2t+3 4t+2
··· ···
Put the labels 3,3,1 to the vertices v ,v ,v respectively. The remaining vertices
2t+2 2t+3 2t+4
SomeNewFamiliesof4-PrimeCordialGraphs 127
v (2t+5 i 4t+2) are labeled as in u (2t+5 i 4t+2).
i i
≤ ≤ ≤ ≤
Case 4. n 3 (mod 4).
≡
Letn=4t+3. Assignthe label2 tothe verticesu (1 i 2t+2). Thenputthe number
i
≤ ≤
3 to the vertices u ,u , ,u . Then assign 1 to the vertices u ,u , ,u
2t+3 2t+5 4t+1 2t+4 2t+6 4t+2
··· ···
and u . Now we turn to the vertices v ,v , ,v . Assign the label 4 to the vertices
4t+3 1 2 4t+3
···
v (1 i 2t+2). The remaining vertices v (2t+3 i 4t+3) are labeled as in u (2t+3
i i i
≤ ≤ ≤ ≤ ≤
i 4t+3). Then relabel the vertex v by 3.
4t+3
≤
The vertex and edge conditions of the above labeling is given in Table 1.
Nature of n v (1) v (2) v (3) v (4) e (0) e (1)
f f f f f f
n 0 (mod 2) n n n n 2n 2 2n 2
≡ 2 2 2 2 − −
n 1 (mod 2) n 1 n+1 n 1 n+1 2n 2 2n 2
≡ −2 2 −2 2 − −
Table 1
It follows that D (P ) is a 4-prime cordial graph for n=2.
2 n
6 2
Theorem 2.2 D (C ) is 4-prime cordial if and only if n 7.
2 n
≥
Proof Let V(D (C )) = u ,v : 1 i n and E(D (C )) = u u , v v , u v ,
2 n i i 2 n i i+1 i i+1 i i+1
{ ≤ ≤ } {
v u : 1 i n 1 u v , v u ,u u ,v v . ClearlyD (C )consistsof2nverticesand
i i+1 n 1 n 1 n 1 n 1 2 n
≤ ≤ − }∪ { }
4n edges. We consider the following cases.
Case 1. n 0 (mod 4).
≡
One can easily check that D (C ) can not havea 4-prime cordiallabeling. Define a vertex
2 4
labeling f from the vertices of D (C ) to the set of first four consecutive positive integers as
2 n
given below.
f(v ) = f(u ) = 2, 1 i n
2i 2i−1 ≤ ≤ 4
f(v ) = f(u ) = 4, 1 i n
2i+1 2i ≤ ≤ 4
f(vn2+2+2i) = f(un2+2+2i) = 1, 1≤i≤ n−44
f(vn2+3+2i) = f(un2+3+2i) = 3, 1≤i≤ n−48
f(un+1)=f(un+2)=f(un+3)=f(vn+2)=3 and f(v1)=f(vn+3)=1.
2 2 2 2 2
Case 2. n 1 (mod 4).
≡
It is easy to verify that D (C ) is not a prime graph. Now we construct a map f :
2 5
V(D (C )) 1,2,3,4 as follows:
2 n
→{ }
f(u ) = 2, 1 i n+3
2i−1 ≤ ≤ 4
f(u ) = 4, 1 i n+3
2i ≤ ≤ 4
f(v ) = 2, 1 i n 1
2i ≤ ≤ −4
f(v ) = 4, 1 i n 1
2i+1 ≤ ≤ −4
f(v ) = f(u ) = 1, 1 i n 5
n+25+2i n+25+2i ≤ ≤ −4
f(v ) = f(u ) = 3, 1 i n 9
n+27+2i n+27+2i ≤ ≤ −4
128 R.Ponraj,RajpalSinghandR.Kala
f(v )=f(v )=f(u )=f(u )=3 and f(v )=f(v )=1.
1 n+5 n+5 n+7 n+7 n+3
2 2 2 2 2
Case 3. n 2 (mod 4).
≡
Obviously D (C ) does not permit a 4-prime cordial labeling. For n = 6, we define a
2 6
6
function f from V(D (C )) to the set 1,2,3,4 by
2 n
{ }
f(u ) = 2, 1 i n+2
2i−1 ≤ ≤ 4
f(u ) = 4, 1 i n+2
2i ≤ ≤ 4
f(v ) = 2, 1 i n 2
2i ≤ ≤ −4
f(v ) = 4, 1 i n 2
2i+1 ≤ ≤ −4
f(v ) = f(u ) = 1, 1 i n 6
n+26+2i n+26+2i ≤ ≤ −4
f(v ) = f(u ) = 3, 1 i n 8
n+28+2i n+28+2i ≤ ≤ −4
and
f(v )=f(v )=f(u )=f(u )=f(u )=3,
1 n+6 n+4 n+6 n+8
2 2 2 2
f(v )=f(v )=f(v )=1.
n+2 n+4 n+8
2 2 2
Case 4. n 3 (mod 4).
≡
Clearly D (C ) is not a 4-prime cordialgraph. Let n=3. Define a map f :V(D (C ))
2 3 2 n
6 →
1,2,3,4 by f(v )=1,
1
{ }
f(v ) = f(u ) = 2, 1 i n+1
2i 2i−1 ≤ ≤ 4
f(v ) = f(u ) = 4, 1 i n+1
2i+1 2i ≤ ≤ 4
f(v ) = f(u ) = 1, 1 i n 3
n+23+2i n+23+2i ≤ ≤ −4
f(v ) = f(u ) = 3, 1 i n 5
n+25+2i n+25+2i ≤ ≤ −4
and f(u )=f(u )=f(v )=3. The Table 2 gives the vertex and edge condition of f.
n+3 n+5 n+5
2 2 2
Nature of n v (1) v (2) v (3) v (4) e (0) e (1)
f f f f f f
n 0,2 (mod 4) n n n n 2n 2n
≡ 2 2 2 2
n 1,3 (mod 4) n 1 n+1 n 1 n+1 2n 2n
≡ −2 2 −2 2
Table 2
It follows that D (C ) is 4-prime cordial iff n 7.
2 n
≥ 2
Example 2.1 A 4-prime cordial labeling of D (C ) is given in Figure 1.
2 9
SomeNewFamiliesof4-PrimeCordialGraphs 129
3
b
2
b
1
b b
2 b
b1 4 b b4
2
1 b b3
4
b
b3 4 2b b2
b
b
3
b
b 4
1
Figure 1
Theorem 2.3 D (K ) is 4-prime cordial if and only if n 0 (mod 2).
2 1,n
≡
Proof It is clear that D (K ) has 2n+2 vertices and 4n edges. Let V(D (K )) =
2 1,n 2 1,n
u,v,u ,v :1 i n and E(D (K ))= uu ,vv ,vu ,uv :1 i n .
i i 2 1,n i i i i
{ ≤ ≤ } { ≤ ≤ }
Case 1. n 0 (mod 2).
≡
Assignthelabel2totheverticesu1,u2, ,un+1. Thenassign4totheverticesun+2, ,
··· 2 2 ···
u ,u,v. Now we move to the vertices v where 1 i n. Assign the label 3 to the vertices
n i
≤ ≤
v (1 i n) then the remaining vertices are labeled with 1. In this case v (1)= v (3) = n,
i ≤ ≤ 2 f f 2
v (2)=v (4)= n +1 and e (0)=e (1)=2n.
f f 2 f f
Case 2. n 1 (mod 2).
≡
Let n = 2t+1. Suppose there exists a 4-prime cordial labeling g, then v (1) = v (2) =
g g
v (3)=v (4)=t+1.
g g
Subcase 2a. g(u)=g(v)=1.
Here e (0)=0, a contradiction.
g
Subcase 2b. g(u)=g(v)=2.
In this case e (0) (t 1)+(t 1)+(t+1)+(t+1)=4t, a contradiction.
g
≤ − −
Subcase 2c. g(u)=g(v)=3.
Then e (0) (t 1)+(t 1)=2t 2, a contradiction.
g
≤ − − −
Subcase 2d. g(u)=g(v)=4.
Similar to Subcase 2b.
Subcase 2e. g(u)=2, g(v)=4 or g(v)=2, g(u)=4.
Here e (0) t+t+t+t=4t, a contradiction.
g
≤
Subcase 2f. g(u)=2, g(v)=3 or g(v)=2, g(u)=3.
Here e (0) (t+1)+t+t=3t+1, a contradiction.
g
≤
130 R.Ponraj,RajpalSinghandR.Kala
Subcase 2g. g(u)=4, g(v)=3 or g(v)=4, g(u)=3.
Similar to Subcase 2f.
Subcase 2h. g(u)=2, g(v)=1 or g(v)=2, g(u)=1.
Similar to Subcase 2f.
Subcase 2i. g(u)=4, g(v)=1 or g(v)=4, g(u)=1.
Similar to Subcase 2h.
Subcase 2j. g(u)=3, g(v)=1 or g(v)=3, g(u)=1.
In this case e (0) t, a contradiction.
g
≤
Hence, if n 1 (mod 2), D (K ) is not a 4-prime cordial graph.
2 1,n
≡ 2
The next investigation is about 4-prime cordial labeling behavior of splitting graph of a
path, star. For a graph G, the splitting graph of G, S (G), is obtained from G by adding for
′
each vertex v of G a new vertex v so that v is adjacent to every vertex that is adjacent to v.
′ ′
Note that if G is a (p,q) graph then S (G) is a (2p,3q) graph.
′
Theorem 2.4 S (P ) is 4-prime cordial for all n.
′ n
Proof Let V(S (P )) = u ,v : 1 i n and E(S (P )) = u u ,u v ,v u : 1
′ n i i ′ n i i+1 i i+1 i i+1
{ ≤ ≤ } { ≤
i n 1 . ClearlyS (P )has2nverticesand3n 3edges. Figure2showsthatS (P ),S (P )
′ n ′ 2 ′ 3
≤ − } −
are 4-prime cordial.
1 3 4 2 3
b b b b b
b b b b b
2 4 2 4 1
Figure 2
For n>3, we consider the following cases.
Case 1. n 0 (mod 4).
≡
We define a function f from the vertices of S (P ) to the set 1,2,3,4 by
′ n
{ }
f(v ) = f(u ) = 2, 1 i n
2i 2i−1 ≤ ≤ 4
f(v ) = f(u ) = 4, 1 i n
2i+1 2i ≤ ≤ 4
f(v ) = f(u ) = 1, 1 i n 4
n+22+2i n+22+2i ≤ ≤ −4
f(v ) = f(u ) = 3, 1 i n 4
n+24+2i n+24+2i ≤ ≤ −4
and f(u )=f(u )=3, f(v )=f(v )=1.
n+2 n+4 1 n+4
2 2 2
In this case v (1)=v (2)=v (3)=v (4)= n, and e (0)= 3n 4, e (1)= 3n 2.
f f f f 2 f 2− f 2−
Case 2. n 1 (mod 4).
≡
SomeNewFamiliesof4-PrimeCordialGraphs 131
We define a map f :V(S (P )) 1,2,3,4 by
′ n
→{ }
f(u ) = 2, 1 i n+3
2i−1 ≤ ≤ 4
f(u ) = 4, 1 i n 1
2i ≤ ≤ −4
f(v ) = 4, 1 i n+3
2i−1 ≤ ≤ 4
f(v ) = 2, 1 i n 1
2i ≤ ≤ −4
f(vn−21+2i) = f(un−21+2i) = 3, 1≤i≤ n−41
f(v ) = f(u ) = 1, 1 i n 1
n+21+2i n+21+2i ≤ ≤ −4
Here v (1)=v (3)= n 1, v (2)=v (4)= n+1, and e (0)=e (1)= 3n 3.
f f −2 f f 2 f f 2−
Case 3. n 2 (mod 4).
≡
Define a vertex labeling f :V(S′(Pn)) 1,2,3,4 by f(v1)=3, f(vn+1)=1,
→{ } 2
f(u ) = 2, 1 i n+2
2i−1 ≤ ≤ 4
f(u ) = 4, 1 i n+2
2i ≤ ≤ 4
f(v ) = 2, 1 i n 2
2i ≤ ≤ −4
f(v ) = 4, 1 i n 2
2i+1 ≤ ≤ −4
f(vn2+2i) = f(un2+2i) = 3, 1≤i≤ n−42
f(v ) = f(u ) = 1, 1 i n 2
n+22+2i n+22+2i ≤ ≤ −4
Here v (1)=v (2)=v (3)=v (4)= n, and e (0)= 3n 4, e (1)= 3n 2.
f f f f 2 f 2− f 2−
Case 4. n 3 (mod 4).
≡
Constructavertexlabelingf fromtheverticesofS (P )totheset 1,2,3,4 byf(u )=1,
′ n n
{ }
f(v )=3,
n
f(v ) = f(u ) = 2, 1 i n+1
2i 2i−1 ≤ ≤ 4
f(v ) = f(u ) = 4, 1 i n+1
2i−1 2i ≤ ≤ 4
f(vn−21+2i) = f(un−21+2i) = 3, 1≤i≤ n−43
f(v ) = f(u ) = 1, 1 i n 3
n+21+2i n+21+2i ≤ ≤ −4
In this case v (1)=v (3)= n 1, v (2)=v (4)= n+1, and e (0)=e (1)= 3n 3.
f f −2 f f 2 f f 2−
Hence S (P ) is 4-prime cordial for all n.
′ n
2
Theorem 2.5 S (K ) is 4-prime cordial for all n.
′ 1,n
Proof Let V(S (K )) = u,v,u ,v : 1 i n and E(S (K )) = uu ,vu ,uv : 1
′ 1,n i i ′ 1,n i i i
{ ≤ ≤ } { ≤
i n . Clearly S (K ) has 2n+2 vertices and 3n edges. The Figure 3 shows that S (K )
′ 1,n ′ 1,2
≤ }
is a 4-prime cordial graph.
132 R.Ponraj,RajpalSinghandR.Kala
1
4 b b
b
2
2 b b
3
b
3
Figure 3
Now for n > 2, we define a map f : V(S (K )) 1,2,3,4 by f(u) = 2, f(v) = 3,
′ 1,n
→ { }
f(u )=1,
n
f(u ) = 2, 1 i n
i ≤ ≤ 2
f(u⌊n2⌋+i) = 3, 1≤i≤(cid:4)n−2(cid:5)1
f(v ) = 4, 1 i (cid:4)n+1(cid:5)
i ≤ ≤ 2
f(v⌈n+21⌉+i) = 1, 1≤i≤(cid:6)n−21(cid:7)
(cid:4) (cid:5)
The Table 3 shows that f is a 4-prime cordial labeling of S (K ).
′ 1,n
2
Values of n v (1) v (2) v (3) v (4) e (0) e (1)
f f f f f f
n 0mod 2 n n +1 n n +1 3n 3n
≡ 2 2 2 2 2 2
n 1mod 2 n+1 n+1 n+1 n+1 3n 1 3n+1
≡ 2 2 2 2 2− 2
Table 3
Next we investigate the 4-prime cordial behavior of degree splitting graph of a star. Let
G=(V,E)beagraphwithV =S S S T whereeachS isasetofverticeshavingat
1 2 t i
∪ ∪···∪ ∪
t
least two vertices and having the same degree and T =V S . The degree splitting graph
i
−
i=1
ofG denotedby DS(G) is obtainedfromGby adding verticeSsw1,w2 ,wt andjoining wi to
···
each vertex of S (1 i t).
i
≤ ≤
Theorem 2.6 DS(B ) is 4-prime cordial if n 1,3 (mod 4).
n,n
≡
Proof Let V(B ) = u,v,u ,v : 1 i n and E(B ) = uv,uu ,vv : 1 i n .
n,n i i n,n i i
{ ≤ ≤ } { ≤ ≤ }
LetV(DS(B ))=V(B ) w ,w andE(DS(B ))=E(B ) w u ,w v ,w u,w v :
n,n n,n 1 2 n,n n,n 1 i 1 i 2 2
∪{ } ∪{
1 i n . Clearly DS(B ) has 2n+4 vertices and 4n+3 edges.
n,n
≤ ≤ }
Case 1. n 1 (mod 4).
≡
Let n = 4t+1. Assign the label 3 to the vertices v ,v , ,v and 1 to the vertices
1 2 2t+1
···
v ,v , ,v . Next assign the label 4 to the vertices u ,u , ,u and 2 to the
2t+2 2t+3 4t+1 1 2 2t+2
··· ···
vertices u ,u , ,u . Finally, assign the labels 1,2,2 and 2 to the vertices w ,u,v
2t+3 2t+4 4t+1 2
···
and w respectively.
1
Case 2. n 3 (mod 4).
≡
As in case 1 assign the labels to the vertices u ,v ,u,v,w and w (1 i n 2). Next
i i 1 2
≤ ≤ −
SomeNewFamiliesof4-PrimeCordialGraphs 133
assign the labels 1,3,2 and 4 respectively to the vertices v ,v ,u and u . The Table 4
n 1 n n 1 n
− −
establishes that this vertex labeling f is a 4-prime cordial labeling.
2
Nature of n v (1) v (2) v (3) v (4) e (0) e (1)
f f f f f f
4t+1 2t+1 2t+2 2t+1 2t+2 8t+3 8t+4
4t+3 2t+2 2t+3 2t+2 2t+3 8t+7 8t+8
Table 4
The final investigation is about 4-prime cordiality of jelly fish graph.
Theorem 2.7 The Jelly fish J(n,n) is 4-prime cordial.
Proof Let V(J(n,n)) = u,v,u ,v ,w ,w : 1 i n and E(J(n,n)) = uu , vu , uw ,
i i 1 2 i i 1
{ ≤ ≤ } {
w v, vw , uw , w w :1 i n . Note that J(n,n) has 2n+4 vertices and 2n+5 edges.
1 2 2 1 2
≤ ≤ }
Case 1. n 0 (mod 4).
≡
Let n = 4t. Assign the label 1 to the vertices u ,u , ,u . Next assign the label 3
1 2 2t+1
···
to the vertices u ,u , ,u . We now move to the other side pendent vertices. Assign
2t+2 2t+3 4t
···
the label3 to the verticesu ,u . Next assignthe label2to the verticesu ,u ,...,u . Then
1 2 3 4 2t+3
assign the label 4 to the remaining pendent vertices. Finally assign the label 4 to the vertices
u,v,w ,w .
1 2
Case 2. n 1 (mod 4).
≡
Letn=4t+1. In this case,assignthe label 1to the vertices v ,v , ,v and3 to the
1 2 2t+1
···
vertices v ,v , ,v . Next assign the label 2 to the vertices u ,u , ,u , and
2t+1 2t+3 4t+1 1 2 2t+2
··· ···
3 to the vertices u and u . Next assign the label 4 to the remaining pendent vertices
2t+3 2t+4
u ,u , ,u . Finally assign the label 4 to the vertices u,v,w ,w .
2t+5 2t+6 4t+1 1 2
···
Case 3. n 2 (mod 4).
≡
As in Case 2, assignthe label to the verticesu ,v (1 i n 1),u,v,w ,w . Next assign
i i 1 2
≤ ≤ −
the labels 1,4 respectively to the vertices u and v .
n n
Case 4. n 3 (mod 4).
≡
Assign the labels to the vertices u,v,w ,w ,u ,v (1 i n 1) as in case 3. Finally
1 2 i i
≤ ≤ −
assignthe labels 2,1respectivelyto the verticesu ,v . The Table 5establishesthatthis vertex
n n
labeling f is obviously a 4-prime cordial labeling.
2
Values of n v (1) v (2) v (3) v (4) e (0) e (1)
f f f f f f
4t 2t+1 2t+1 2t+1 2t+1 4t+3 4t+2
4t+1 2t+1 2t+2 2t+2 2t+1 4t+4 4t+3
4t+2 2t+2 2t+2 2t+2 2t+2 4t+5 4t+4
4t+3 2t+3 2t+3 2t+2 2t+2 4t+6 4t+5
Table 5
Corollary 2.1 The Jelly fish J(m,n) where m n is 4-prime cordial.
≥
134 R.Ponraj,RajpalSinghandR.Kala
Proof Let m=n+r, r 0. Use of the labeling f given in theorem ?? assign the label to
≥
the vertices u,v,w ,w ,u ,v (1 i n).
1 2 i i
≤ ≤
Case 1. r 0 (mod 4).
≡
Let r =4k, k N. Assignthe label 2 to the vertices u ,u , ,u andto the ver-
n+1 n+2 n+k
∈ ···
ticesu ,u , ,u . Thenassignthelabel1totheverticesu ,u , ,u
n+k+1 n+k+2 n+2k n+2k+1 n+2k+2 n+3k
··· ···
and 3 to the vertices u , u , ..., u . Clearly this vertex labeling is a 4-prime
n+3k+1 n+3k+2 n+4k
cordial labeling.
Case 2. r 1 (mod 4).
≡
Let r =4k+1, k N. Assign the labels to the vertices u (1 i r 1) as in case 1.
n+i
∈ ≤ ≤ −
If n 0,1,2 (mod 4), then assign the label 1 to the vertex u ; otherwise assign the label 4 to
r
≡
the vertex u .
r
Case 3. r 2 (mod 4).
≡
Let r =4k+2, k N. As in Case 2 assignthe labels to the vertices u (1 i r 1).
n+i
∈ ≤ ≤ −
Then assign the label 4 to the vertex u .
r
Case 4. r 3 (mod 4).
≡
Let r = 4k+3, k N. In this case assign the label 3 to the last vertex and assign the
∈
label to the vertices u (1 i r 1) as in Case 3.
n+i
≤ ≤ − 2
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