Path Related n-Cap Cordial Graphs

2017·0.14 MB·

by  A. Nellai Murugan

International J.Math. Combin. Vol.3(2017), 119-124
Path Related n-Cap Cordial Graphs
A. Nellai Murugan
(DepartmentofMathematics,V.O.ChidambaramCollege,TamilNadu,India)
P. Iyadurai Selvaraj
(DepartmentofComputerScience,V.O.ChidambaramCollege,TamilNadu,India)
E-mail: [email protected],[email protected]
Abstract: Let G = (V,E) be a graph with p vertices and q edges. A n-cap ( ) cordial
labeling ofagraphGwithvertexsetVisabijection from Vto 0,1 suchthatifeach edge
{ } V
uv is assigned thelabel
0, if f(u)=f(v)=1
f(uv)=
1, otherwise.

with the condition that the number of vertices labeled with 0 and the number of vertices

labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number
of edges labeled with 1 differ by at most 1. The graph that admits a cordial labeling is
called a cordial graph (nCCG). Inthispaper,weprovedthatPath PVn, Comb (Pn K1),
⊙
Pm 2KV1 and Fan (Fn =Pn+K1) are cordial graphs.
⊙
Key Words: cordiallabeling,SmaraVndachelycordiallabeling, cordiallabelinggraph.
AMS(2010):V05C78. V
§1. Introduction
A graph G is a finite non-empty set of objects called vertices together with a set of unordered
pairsofdistinctverticesofGwhichiscallededges. Eachpaire= uv ofverticesinE iscalled
{ }
an edge or a line of G. In this paper, we proved that Path P , Comb (P ), P 2K and
n n 1 m 1
⊙ ⊙
Fan (F =P +K ) are cordial graphs.
n n 1
V
§2. Preliminaries
LetG=(V,E) be agraphwith p verticesandq edges. An-cap ( ) cordiallabeling ofagraph
G with vertex set V is a bijection from V to 0,1 such that if each edge uv is assigned the
{ } V
1ReceivedSeptember 30,2016,Accepted August26,2017.
120 A.NellaiMuruganandP.IyaduraiSelvaraj
label
0, if f(u)=f(v)=1
f(uv)=
1, otherwise.

withtheconditionthatthenumberofverticeslabeledwith0andthenumberofverticeslabeled
with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges
labeled with 1 differ by at most 1, and it is said to be a Smarandachely cordial labeling if the
number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at least 1
andthe number ofedges labeledwith 0 orthe number ofedges labeledwith 1 differ by at least
1.
The graph that admits a cordial labeling is called a cordial graph. We proved that
Path P , Comb (P K ), P 2K and Fan (F =P +K ) are cordial graphs.
n n⊙ 1 mV⊙ 1 n n V1
V
Definition 2.1 A path is a graph with sequence of vertices u , u , , u such that successive
1 2 n
···
vertices are joined with an edge, denoted by P , which is a path of length n 1.
n
−
A closed path of length n is cycle C .
n
Definition 2.2 A comb is a graph obtained from a path P by joining a pendent vertex to each
n
vertices of P , it is denoted by P K
n n 1
⊙
Definition 2.3 A graph obtained from a path P by joining two pendent vertices at each
m
vertices of P is denoted by P 2K
m m 1
⊙
Definition 2.4 A fan is a graph obtained from a path P by joining each vertices of P to a
n n
pendent vertex, it is denoted by F =P +K
n n 1
§3. Main Results
Theorem 3.1 A path P is a cordial graph
n
V
Proof Let V(P ) = u :1 i n and E(P ) = u u :1 i n 1 Define f :
n i n i i+1
{ ≤ ≤ } { ≤ ≤ − }
V(P ) 0,1 with the vertex labeling determined following.
n
→{ }
Case 1. n is odd.
Define
0, 1 i n 1,
f(u )= ≤ ≤ −2
i 1, n+1 i n.
 2 ≤ ≤
The induced edge labeling are 
1, 1 i n,
f∗(uiui+1)=0, n≤ i≤ 2n.
 2 ≤ ≤
HereV (f)+1=V (f)ande (f)=e (f). Clearly,itsatisfiesthecondition V (f) V (f) 1
0 1 0 1 0 1
| − |≤
and e (f) e (f) 1.
0 1
| − |≤
PathRelatedn-CapCordialGraphs 121
Case 2. n is even.
Define
0, 1 i n,
f(u )= ≤ ≤ 2
i 1, n +1 i n.
 2 ≤ ≤
The induced edge labeling are 
1, 1 i n,
f∗(uiui+1)=0, n≤+1≤ 2i n.
 2 ≤ ≤
HereV (f)=V (f)ande (f)+1=e (f)whichsatisfiesthe condition V (f) V (f) 1and
0 1 0 1 0 1
| − |≤
e (f) e (f) 1. Hence, a path P is a cordial graph.
0 1 n
| − |≤ 2
V
For example, P and P are cordial graph shown in the Figure 1.
5 6
V
u 1 u 1 u 0 u 0 u
1 2 3 4 5
0 0 1 1 1
u 1 u 1 u 1 u 0 u 0 u
1 2 3 4 5 6
0 0 0 1 1 1
Figure 1
Theorem 3.2 A comb P K is a cordial graph
n 1
⊙
V
Proof Let G be a comb P K and let V(G) = (u ,v ) : 1 i n and E(G) =
n 1 i i
⊙ { ≤ ≤ }
[(u u ) : 1 i n 1] [(u v ) : 1 i n] . Define f : V(G) 0,1 with a vertex
i i+1 i i
{ ≤ ≤ − ≤ ≤ } → { }
labeling
S
f(u ) = 1, 1 i n,
i
≤ ≤
f(v ) = 0, 1 i n.
i
≤ ≤
The induced edge labeling are
f (u u ) = 1, 1 i<n,
∗ i i+1
≤
f (u v ) = 0, 1 i n.
∗ i i
≤ ≤
HereV (f)=V (f)ande (f)=e (f)+1whichsatisfiesthe condition V (f) V (f) 1and
0 1 0 1 0 1
| − |≤
e (f) e (f) 1. Hence, a comb P K is a cordial graph.
0 1 n 1
| − |≤ ⊙ 2
V
For example, P K is a cordial graph shown in Figure 2.
5 1
⊙
V
122 A.NellaiMuruganandP.IyaduraiSelvaraj
1 1 1 1 1
u 0 u 0 u 0 u 0 u
1 2 3 4 5
1 1 1 1 1
v v v v v
1 2 3 4 5
0 0 0 0 0
Figure 2
Theorem 3.3 A graph P 2K is a cordial graph.
m 1
⊙
V
Proof Let G be a P 2K with V(G)= u ,v ,v ,1 i n and E(G)= [(u u ):
m 1 i 1i 2i i i+1
⊙ { ≤ ≤ } {
1 i<n] [(u v ):1 i n] [(u v ):1 i n] . Define f :V(C ) 0,1 by a vertex
i 1i i 2i n
≤ ≤ ≤ ≤ ≤ } →{ }
labeling f(u )= 1,1 i n , f(v )= 0,1 i n and if n is even,
Si { ≤ ≤ } S 1i { ≤ ≤ }
1, 1 i n,
f(v )= ≤ ≤ 2
2i 0, n +1 i n,
 2 ≤ ≤

if n is odd
1, 1 i n+1,
f(v )= ≤ ≤ 2
2i 0, n+1 +1 i n.
 2 ≤ ≤
The induced edge labeling are 
f (u u ) = 0,1 i n ,
∗ i i+1
{ ≤ ≤ }
f∗(uiv1i) = 1,1 i n
{ ≤ ≤ }
and if n is even
0, 1 i n,
f (u v )= ≤ ≤ 2
∗ i 2i 1, n +1 i n.
 2 ≤ ≤
HereV0(f)=V1(f)ande0(f)+1=e1(f)whichsatisfiesthe condition V0(f) V1(f) 1and
| − |≤
e (f) e (f) 1, and if n is odd
0 1
| − |≤
0, 1 i n+1,
f∗(uiv2i)=1, n+≤1 ≤i 2n.
 2 ≤ ≤
HereV (f)+1=V (f)ande (f)=e (f)whichsatisfiesthe condition V (f) V (f) 1and
0 1 0 1 0 1
| − |≤
e (f) e (f) 1. Hence, P 2K is a cordial graph.
0 1 m 1
| − |≤ ⊙ 2
V
For example, P 2K is a cordial graph shown in the Figures 3.
5 1
⊙
V
PathRelatedn-CapCordialGraphs 123
1 1 1 0 0
v v v v v
21 22 23 24 25
0 0 0 1 1
u 0 u 0 u 0 u 0 u
1 2 3 4 5
1 1 1 1 1
1 1 1 1 1
v v v v v
11 12 13 14 15
0 0 0 0 0
Figure 3
Theorem 3.4 A fan F =P +K is a cordial graph if n is even.
n n 1
V
Proof Let G be a fan F = P +K and n is even with V(G) = u,v : 1 6 i 6 n and
n n 1 i
{ }
E(G)= (u,v ):16i6n . Define f :V(G) 0,1 with a vertex labeling f(u)= 1 and
i
{ } →{ } { }
1, 1 i n,
f(v )= ≤ ≤ 2
i 0, n +1 i n.
 2 ≤ ≤
The induced edge labeling are 
0, 1 i n, 0, 1 i n,
f∗(uvi)= ≤ ≤ 2 and f∗(vivi+1)= ≤ ≤ 2
1, n +1 i n, 1, n i n.
2 ≤ ≤ 2 ≤ ≤
HereV (f)+1=V (f)ande (f)+1=e (f)whichsatisfiestheconditions V (f) V (f) 1
0 1 0 1 0 1
| − |≤
and e (f) e (f) 1. Hence, a fan F =P +K is a cordial graph if n is even.
0 1 n n 1
| − |≤ 2
For example, a fan F =P +K is cordial shownVin Figure 4.
6 6 1
V
u
0
0 1 1
0 1
v 0 v 0 v 1 v 1 v 1 v
1 2 3 4 5 6
1 1 1 0 0 0
Figure 4
124 A.NellaiMuruganandP.IyaduraiSelvaraj
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