Cordiality in the Context of Duplication in Web and Armed Helm

2017·0.17 MB·

by  U. M. Prajapati

International J.Math. Combin. Vol.3(2017), 90-105
Cordiality in the Context of Duplication in Web and Armed Helm
U M Prajapati
(St. Xavier’sCollege,Ahmedabad-380009,Gujarat,India)
R M Gajjar
(DepartmentofMathematics,SchoolofSciences,GujaratUniversity,Ahmedabad-380009,Gujarat,India)
E-mail: [email protected],[email protected]
Abstract: Let G = (V(G),E(G)) be a graph and let f : V(G) 0,1 be a mapping
→ { }
from the set of vertices to 0,1 and for each edge uv E assign the label f(u) f(v).
{ } ∈ | − |
If the numberof vertices labeled with 0 and the numberof vertices labeled with 1 differ by
at most 1 and the number of edges labled with 0 and the number of edges labeled with 1
differbyat most 1, thenf iscalled acordial labeling. Wediscusscordial labeling of graphs
obtained from duplication of certain graph elements in web and armed helm.
Key Words: Graphlabeling,cordiallabeling,cordialgraph,Smarandachelycordiallabel-
ing, Smarandachely cordial graph.
AMS(2010): 05C78
§1. Introduction
We begin with simple, finite, undirected graph G = (V(G),E(G)) where V(G) and E(G)
denotes the vertex set and the edge set respectively. For all other terminology we follow West
[1]. We will give the brief summary of definitions which are useful for the present work.
Definition 1.1 The graph labeling is an assignment of numbers to the vertices or edges or both
subject to certain condition(s).
A detailed survey of various graph labeling is explained in Gallian [3].
Definition 1.2 For a graph G = (V(G),E(G)), a mapping f : V(G) 0,1 is called a
→ { }
binary vertex labeling of G and f(v) is called the label of the vertex v of G under f. For an
edge e=uv, the induced edge labeling f :E(G) 0,1 defined as f (uv)= f(u) f(v).
∗ ∗
→{ } | − |
Let v (0),v (1) be the number of vertices of G having labels 0 and 1 respectively under f
f f
and let e (0),e (1) be the number of edges having labels 0 and 1 respectively under f .
f f ∗
Definition 1.3 Duplication of a vertex v of a graph G produces a new graph G by adding a
′
new vertex v such that N(v )=N(v). In other words a vertex v is said to be duplication of v
′ ′ ′
if all the vertices which are adjacent to v in G are also adjacent to v in G.
′ ′
Definition 1.4 Duplication of an edge e=uv of a graph G produces a new graph G by adding
′
1ReceivedDecember19,2016,Accepted August24,2017.
CordialityintheContextofDuplicationinWebandArmedHelm 91
an edge e =uv such that N(u)=N(u) v v and N(v )=N(v) u u .
′ ′ ′ ′ ′ ′ ′
∪{ }−{ } ∪{ }−{ }
Definition 1.5 The wheel W , is join of the graphs C and K . i.e W = C +K . Here
n n 1 n n 1
vertices corresponding to C are called rim vertices and C is called rim of W while, the vertex
n n n
corresponding to K is called the apex vertex, edges joining the apex vertex and a rim vertex is
1
called spoke.
Definition 1.6([3])The helm H , is the graph obtained from the wheel W by adding a pendant
n n
edge at each rim vertex.
Definition1.7([3]) ThewebWb , is thegraph obtained byjoiningthependentpoints of ahelm
n
to form a cycle and then adding a single pendent edge to each vertex of this outer cycle, here
verticescorrespondingtothisoutercyclearecalledouterrimverticesandverticescorresponding
to wheel except the apex vertex are called inner rim vertices.
We define one new graph family as follows:
Definition 1.8 An armed helm is a graph in which path P is attached at each vertex of wheel
2
W by an edge. It is denoted by AH where n is the number of vertices in cycle C .
n n n
Definition 1.9 A binary vertex labeling f of a graph G is called a cordial labeling if v (1)
f
| −
v (0) 1 and e (1) e (0) 1, and a binary vertex labeling f of a graph G is called a
f f f
| ≤ | − | ≤
Smarandachely cordial labeling if v (1) v (0) 1 or e (1) e (0) 1.
f f f f
| − |≥ | − |≥
A graph G is said to be cordial if it admits cordial labeling, and Smarandachely cordial if
it admits Smarandachely cordial labeling.
The concept of cordial labeling was introduced by Cahit [2] in which he proved that the
wheel W is cordial if and only if n 3(mod4). Vaidya and Dani [4] proved that the graphs
n
6≡
obtained by duplication of an arbitrary edge of a cycle and a wheel admit a cordial labeling.
Prajapati and Gajjar [5] proved that complement of wheel graph and complement of cycle
grapharecordialif n 4(mod8) or n 7(mod8). PrajapatiandGajjar [6]provedthat cordial
6≡ 6≡
labeling in the context of duplication of cycle graph and path graph.
§2. Main Results
Theorem 2.1 The graph obtained by duplicating all the vertices of the web Wb is cordial.
n
Proof Let V(Wb )= t u ,v ,w ,/1 i n and E(Wb )= tu ,u v ,v w /1 i
n i i i n i i i i i
{ } { ≤ ≤ } { ≤ ≤
n u u ,v v u u ,v v /1 i n 1 . LetGbethegraphobtainedbyduplicating
} { n 1 n 1} { i i+1 iSi+1 ≤ ≤ − }
all the vertices in Wb . Let t,u ,u , ,u ,v ,v , ,v ,w ,w , ,w be the new vertices
S S n ′ ′1 ′2 ··· ′n 1′ 2′ ··· n′ 1′ 2′ ··· n′
of G by duplicating t,u ,u , ,u ,v ,v , ,v ,w ,w , ,w respectively. Then V(G) =
1 2 n 1 2 n 1 2 n
··· ··· ···
t,t u ,v ,w ,u ,v ,w /1 i n andE(G)= tu ,u v ,v w ,w v ,u v ,u v ,tu ,v w ,tu
{ ′}∪{ i i i ′i i′ i′ ≤ ≤ } { i i i i i i i′ i i′ ′i i ′i i i′ ′ i
/1 i n u u ,v v ,v v ,v v ,u u ,u u u u ,v v ,v v ,v v ,u u ,u u
≤ ≤ } { n 1 n 1 n′ 1 n 1′ ′n 1 n ′1} { i i+1 i i+1 i′ i+1 i i′+1 ′i i+1 i ′i+1
/1 i n 1 . Therefore V(G) = 6n+2 and E(G) = 15. Using parity of n, we have the
≤ ≤ −S } | | |S |
following cases:
Case 1. n is even.
92 UMPrajapatiandRMGajjar
Define a vertex labeling f :V(G) 0,1 as follows:
→{ }
1 if x=t;
′
1 if x=w ,i 2,4,...,n 2,n ;
f(x)=110 iiifff xxx=∈={wt;uii′i,,vi∈i′∈},{{i1,∈3,{.1.,.2,,n.−.−.,3n,n−}−1,1n}};;
Thus v (1)=3n+1 andv000(0)iiifff=xxx3∈==n{ww+vii′i,,1,.uii′iT∈∈}h,{{ei21i,,∈n43d,,{u..1..c,..e2,,d,nn.e.−−d.g,23ne,,lnn−a}b−1;e,l1ni}n}.g; f :E(G) 0,1 is
f f ∗
→{ }
f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e tu ,u v , i 1,2,...,n 1,n ;
i i i
∈{ } ∈{ − }
0 if e u u ,v v , i 1,2,...,n 2,n 1 ;
i i+1 i i+1
f∗(e)=1001 iiiiffff eeee∈∈∈∈∈{{{{{utvv′iiiuwwui′iii,′′+,,twwu1,′iii,uvvuii′i′′u}}′iv,,i+i,ii1u}∈∈,ivvi{{i′v12}∈i′,,,+341{i,,,..∈v..i′..v{,,i1nn+,12−−},,.32.,,i.−nn,∈}n−;{−11,}1−2;,,n..}}.;,n−2,n−1};
1 if e=v w , i 1,3,...,n 3,n 1 ;
i i
010 iiifff eee=∈∈{{vuuiw′nnuui,11,,iuv∈∈nnvu{{1′12},,;v4n′,v.1..,,vnnv−−1′}2.,n}−; }
15n 15n
Thus e (1)= and e (0)= .
f f
2 2
Case 2 n is odd.
Define a vertex labeling f :V(G) 0,1 as follows:
→{ }
1 if x=t;
′
f(x)=1110 iiiiffff xxxx==∈={wwt;uii′i,,,iiv∈∈i′},{{i21,,∈43,,{..1..,..2,,,nn..−−.,32n,,nn−}−1;,1n}};;
000 iiifff xxx∈=={wwvii′i,,,uii′i∈∈},{{i21,,∈43,,{..1..,..2,,,nn..−−.,32n,,nn−}−1.,1n}};;
CordialityintheContextofDuplicationinWebandArmedHelm 93
Thus v (1)=3n+1 and v (0)=3n+1. The induced edge labeling f :E(G) 0,1 is
f f ∗
→{ }
f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e tu ,u v , i 1,2,...,n 1,n ;
i i i
∈{ } ∈{ − }
0 if e u u ,v v , i 1,2,...,n 2,n 1 ;
i i+1 i i+1
f∗(e)=1001 iiiiffff eeee∈∈∈∈∈{{{{{utvv′iiiuwwui′iii,′′+,,twwu1,′iii,uvvuii′i′′u}}′iv,,i+i,ii1u}∈∈,ivvi{{i′v12}∈i′,,,+341{i,,,..∈v..i′..v{,,i1nn+,12−−},,.23.,,i.−nn,∈}n−;{−11,}1−2;,,n..}}.;,n−2,n−1};
1 if e=v w , i 1,3,...,n 2,n ;
i i
001 iiifff eee=∈∈{{vuuiwn′nuui,11,,ivu∈∈nnvu{{1′12},,;v4n′,v.1..,,vnnv−−1′}3.,n}−1};
15n 1 15n+1
Thus e (1)= − and e (0)= .
f f
2 2
From both the cases we can conclude v (1) v (0) 1 and e (1) e (0) 1. So, f
f f f f
| − | ≤ | − | ≤
admits cordial labeling on G. Hence G is cordial.
2
Theorem 2.2 The graph obtained by duplicating all the pendent vertices of the web Wb is
n
cordial.
Proof Let V(Wb ) = t u ,v ,w /1 i n and E(Wb ) = tu ,u v ,v w /1
n i i i n i i i i i
{ } { ≤ ≤ } { ≤
i n u u ,v v u u ,v v /1 i n 1 . Let G be the graph obtained by
≤ } ∪ { n 1 n 1} { i i+1S i i+1 ≤ ≤ − }
duplicating all the pendent vertices in Wb . Let w ,w , ,w be the new vertices of G by
S n 1′ 2′ ··· n′
duplicating w ,w , ,w respectively. Then V(G) = t u ,v ,w ,w /1 i n and
1 2 ··· n { }∪{ i i i i′ ≤ ≤ }
E(G) = tu ,u v ,v w ,v w /1 i n u u ,v v u u ,v v /1 i n 1 .
{ i i i i i i i′ ≤ ≤ } ∪ { n 1 n 1} { i i+1 i i+1 ≤ ≤ − }
Therefore V(G) = 4n+1 and E(G) = 6n. Define a vertex labeling f : V(G) 0,1 as
| | | | S → { }
follows:
0 if x=t;
f(x)=1 if x w ,u , i 1,2,...,n 1,n ;
i i
0 if x∈{v ,w }, i ∈{1,2,...,n −1,n}.
∈{ i i′} ∈{ − }
Thus vf(1) = 2n and vf(0) = 2n+1. The induced edge labeling f∗ : E(G) 0,1 is
→ { }
f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e tu ,u v ,v w , i 1,2,...,n 1,n ;
i i i i i
∈{ } ∈{ − }
0 if e=v w , i 1,2,...,n 1,n ;
f∗(e)=0 if e∈{uiiui′i+1,∈vi{vi+1}, i∈{−1,2,.}..,n−2,n−1};
0 if e u u ,v v .
n 1 n 1
 ∈{ }
Thus e (1) = 3n and e (0) = 3n. Therefore f satisfies the conditions v (1) v (0) 1 and
f f f f
| − | ≤
94 UMPrajapatiandRMGajjar
e (1) e (0) 1. So, f admits cordial labeling on G. Hence G is cordial.
f f
| − |≤ 2
Theorem 2.3 The graph obtained by duplicating the outer rim vertices and the apex of the web
Wb is cordial.
n
Proof Let V(Wb )= t u ,v ,w ,/1 i n and E(Wb )= tu ,u v ,v w /1 i
n i i i n i i i i i
{ } { ≤ ≤ } { ≤ ≤
n u u ,v v /1 i n 1 u u ,v v . LetGbethegraphobtainedbyduplicating
} { i i+1 i i+1 ≤ ≤ −S } { n 1 n 1}
the outer rim vertices and the apex in Wb . Let t,v ,v , ,v be the new vertices of G
S S n ′ 1′ 2′ ··· n′
by duplicating t,v ,v , ,v respectively. Then V(G) = t,t u ,v ,w ,v /1 i n
1 2 ··· n { ′} { i i i i′ ≤ ≤ }
and E(G) = tu ,u v ,v w ,w v ,u v ,tu /1 i n u u ,v v ,v v ,v v /1 i
{ i i i i i i i′ i i′ ′ i ≤ ≤ } { i i+1 iSi+1 i′ i+1 i i′+1 ≤ ≤
n 1 u u ,v v ,v v ,v v . Therefore V(G) =4n+2and E(G) =10n. Defineavertex
− } { n 1 n 1 n′ 1 n 1′} | | S | |
labeling f :V(G) 0,1 as follows:
S →{ }
0 if x=t;
1 if x u ,w , i 1,2,...,n 1,n ;
i i
f(x)=0 if x∈∈{{vi,vi′}}, i∈∈{{1,2,...,n−−1,n}};
1 if x=t.
′

Thus vf(1)=2n+1 and vf(0)=2n+1. The induced edge labeling f∗ :E(G) 0,1 is
→{ }
f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e tu ,u v ,v w ,w v ,u v , i 1,2,...,n 1,n ;
∈{ i i i i i i i′ i i′} ∈{ − }
0 if e u u ,v v ,v v ,v v , i 1,2,...,n 2,n 1 ;
f∗(e)=0 if e∈={t′uii,i+i1∈{i1,i+2,1...i′,in+−1 1i,ni′+};1} ∈{ − − }
0 if e u u ,v v .v v ,v v .
 ∈{ n 1 n 1 n′ 1 n 1′}
Thus e (1) = 5n and e (0) = 5n. Therefore f satisfies the conditions v (1) v (0) 1 and
f f f f
| − | ≤
e (1) e (0) 1. So, f admits cordial labeling on G. Hence G is cordial.
f f
| − |≤ 2
Theorem 2.4 The graph obtained by duplicating all the vertices except the apex vertex of the
web Wb is cordial.
n
Proof Let V(Wb ) = t u ,v ,w /1 i n and E(Wb ) = tu ,u v ,v w /1 i
n i i i n i i i i i
{ } { ≤ ≤ } { ≤ ≤
n u u ,v v u u ,v v /1 i n 1 . LetGbethegraphobtainedbyduplicating
} { n 1 n 1} { i i+1 iSi+1 ≤ ≤ − }
alltheverticesexcepttheapexvertexinWb . Letu ,u , ,u ,v ,v , ,v ,w ,w , ,w
S S n ′1 ′2 ··· ′n 1′ 2′ ··· n′ 1′ 2′ ··· n′
be the new vertices of G by duplicating u ,u , ,u ,v ,v , ,v ,w ,w , ,w respec-
1 2 n 1 2 n 1 2 n
··· ··· ···
tively. ThenV(G)= t u ,v ,w ,u ,v ,w /1 i n andE(G)= tu ,u v ,v w ,w v ,u v ,
{ } { i i i ′i i′ i′ ≤ ≤ } { i i i i i i i′ i i′
u v ,v w ,tu /1 i n u u ,v v ,v v ,v v ,u u ,u u u u ,v v ,v v ,v v ,
′i i i i′ ′i ≤ ≤ }S{ n 1 n 1 n′ 1 n 1′ ′n 1 n ′1} { i i+1 i i+1 i′ i+1 i i′+1
u u ,u u /1 i n 1 . Therefore V(G) = 6n+1 and E(G) = 14n. Define a vertex
′i i+1 i ′i+1 ≤ ≤ S− } | | S| |
CordialityintheContextofDuplicationinWebandArmedHelm 95
labeling f :V(G) 0,1 as follows:
→{ }
0 if x=t;
f(x)=1 if x w ,u ,u , i 1,2,...,n 1,n ;
0 if x∈{vi,wi,v′i}, i ∈{1,2,...,n −1,n}.
∈{ i i′ i′} ∈{ − }

Thus vf(1) = 3n and vf(0) = 3n+1. The induced edge labeling f∗ : E(G) 0,1 is
→ { }
f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e tu ,u v ,v w ,w v ,u v ,u v ,tu , i 1,2,...,n 1,n ;
∈{ i i i i i i i′ i i′ ′i i ′i} ∈{ − }
0 if e=v w , i 1,2,...,n 1,n ;
f∗(e)=0 if e∈{uiiui′i+1,∈vi{vi+1,vi′vi+1−,vivi′+}1,u′iui+1,uiu′i+1}, i∈{1,2,...,n−2,n−1};
0 if e u u ,v v .v v ,v v ,u u ,u u .
 ∈{ n 1 n 1 n′ 1 n 1′ ′n 1 n ′1}
Thus e (1) = 7n and e (0) = 7n. Therefore f satisfies the conditions v (1) v (0) 1 and
f f f f
| − | ≤
e (1) e (0) 1. So, f admits cordial labeling on G. Hence G is cordial.
f f
| − |≤ 2
Theorem 2.5 The graph obtained by duplicating all the inner rim vertices and the apex vertex
of the web Wb is cordial.
n
Proof Let V(Wb )= t u ,v ,w ,/1 i n and E(Wb )= tu ,u v ,v w /1 i
n i i i n i i i i i
{ } { ≤ ≤ } { ≤ ≤
n u u ,v v u u ,v v /1 i n 1 . LetGbethegraphobtainedbyduplicating
}∪{ n 1 n 1} { i i+1 iSi+1 ≤ ≤ − }
allthe innerrimverticesandthe apexvertexinWb . Lett,u ,u , ,u be the newvertices
S n ′ ′1 ′2 ··· ′n
ofGbyduplicatingt,u ,u , ,u respectively. ThenV(G)= t,t u ,v ,w ,u /1 i n
1 2 ··· n { ′}∪{ i i i ′i ≤ ≤ }
and E(G) = tu ,u v ,v w ,u v ,tu ,tu /1 i n u u ,v v ,u u ,u u /1 i
{ i i i i i ′i i ′i ′ i ≤ ≤ }∪{ i i+1 i i+1 ′i i+1 i ′i+1 ≤ ≤
n 1 u u ,v v ,u u ,u u . Therefore V(G) = 4n+2 and E(G) = 10n. Define a
− }∪{ n 1 n 1 ′n 1 n ′1} | | | |
vertex labeling f :V(G) 0,1 as follows:
→{ }
0 if x=t;
1 if x w ,u , i 1,2,...,n 1,n ;
i i
f(x)=0 if x∈∈{{vi,u′i}}, i∈∈{{1,2,...,n−−1,n}};
1 if x=t.
′

Thus v (1)=2n+1 and v (0)=2n+1. The induced edge labeling f :E(G) 0,1 is
f f ∗
→{ }
f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e tu ,u v ,v w , i 1,2,...,n 1,n ;
i i i i i
∈{ } ∈{ − }
1 if e u u ,u u , i 1,2,...,n 2,n 1 ;
f∗(e)=0 if e∈∈{{u′iiuii++11,viivi′i++11}}, i∈∈{{1,2,...,n−−2,n−−1}};
0 if e tu ,tu ,u v , i 1,2,...,n 1,n
 ∈{ ′i ′ i ′i i} ∈{ − }
96 UMPrajapatiandRMGajjar
and
0 if e u u ,v v ;
n 1 n 1
f (e)= ∈{ }
∗
1 if e u u ,u u .
 ∈{ ′n 1 n ′1}
Thus ef(1) = 5n and ef(0) = 5n. Therefore f satisfies the conditions vf(1) vf(0) 1 and
| − | ≤
e (1) e (0) 1. So, f admits cordial labeling on G. Hence G is cordial.
f f
| − |≤ 2
Theorem 2.6 The graph obtained by duplicating all the edges other than spoke edges of the
web Wb is cordial.
n
Proof Let V(Wb )= t u ,v ,w /1 i n and E(Wb )= j =tu ,l =u v ,o =
n i i i n i i i i i i
{ } { ≤ ≤ } {
v w /1 i n k =u u ,m =v v /1 i n 1 k =u u ,m =v v . Let G
i i ≤ ≤ } { i i i+1S i i i+1 ≤ ≤ − } { n n 1 n n 1}
bethegraphobtainedbyduplicatingalltheedgesotherthanspokeedgesinWb . Foreachi
S S n ∈
1,2, ,n,letk =a b ,l =c d ,m =e f ando =g h be thenewedgesofGbyduplicating
··· i′ i i i′ i i ′i i i ′i i i
k ,l ,m and o respectively. Then V(G) = t u ,v ,w ,a ,b ,c ,d ,e ,f ,g ,h /1 i n
i i i i i i i i i i i i i i i
{ }∪{ ≤ ≤ }
andE(G)= b u ,v f /1 i n 2 tu ,u v ,a b ,e f ,c d ,g h ,tc ,tb ,g u ,ta ,a v ,
i i+2 i i+2 i i i i i i i i i i i i i i i i i i
{ ≤ ≤ − } {
e u ,d w ,v w ,e w /1 i n b v ,u a ,f u ,v e ,u u ,v v ,d v ,v d ,
i i i i i i i i ≤ ≤ } { i i+1Si i+1 i i+1 i i+1 i i+1 i i+1 i i+1 i i+1
c u ,u c ,g v ,v g ,f w /1 i n 1 u u ,v v ,d v ,v d ,c u ,u c ,g v ,
i i+1 i i+1 i i+1 i i+1 i i+1S ≤ ≤ − } { n 1 n 1 n 1 n 1 n 1 n 1 n 1
v g ,b u ,b u ,v f ,f v ,b v ,u a ,f u ,v e ,f w . Therefore V(G) =11n+1 and
n 1 n−1 1 n 2 n−1 1 n 2 n 1 n 1 n 1 n S1 n 1} | |
E(G) =30n. Using parity of n, we have the following cases:
| |
Case 1 n is even.
Define a vertex labeling f :V(G) 0,1 as follows:
→{ }
0 if x=t;
0 if x v ,a ,d ,f ,g , i 1,2,...,n 1,n ;
i i i i i
f(x)=11 iiff xx∈∈={{wui,,bii,ci,1e,i3,,h.i.}}.,,ni∈∈{{31,,n2,..1.,;n−−1,n}};
i
∈{ − − }
11n 0 if x=1w1in, i∈{2,4,...,n−2,n}.
Thus vf(1)= and vf(0)= +1. The induced edge labeling f∗ :E(G) 0,1 is
2 2 →{ }
f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e tu ,u v ,a b ,e f ,c d ,g h ,tc ,tb ,g u , i 1,2,...,n 1,n ;
i i i i i i i i i i i i i i i
∈{ } ∈{ − }
0 if e ta ,a v ,e u , i 1,2,...,n 1,n ;
i i i i i
f∗(e)=01 iiff ee∈∈∈{{{ubiivuii++11,,uviiavii++11},,fdiiuv∈ii++{11,,vviiedii++11},,ci−iui∈+1{,1u},i2c,i.+.1.,,gniv−i+21,,nvi−gi+11}};,1≤i≤n−1;
0 if e b u ,v f , i 1,2,...,n 3,n 2 ;
i i+2 i i+2
1 if e∈∈{{diwi,viwi}, i}∈{1∈,3{,...,n−3,−n−1}− }
CordialityintheContextofDuplicationinWebandArmedHelm 97
and
0 if e d w ,v w , i 2,4,...,n 2,n ;
i i i i
∈{ } ∈{ − }
0 if e e w ,f w , i 1,3,...,n 3,n 1 ;
i i i i+1
f∗(e)=11 iiff ee∈=={efiiwwii+,1i,∈i{∈2,{}42,,.4.,∈..,.n{.,−n2−,n4,}n; −−2}; − }
0 if e u u ,v v ,d v ,v d ,c u ,u c ,g v ,v g ,b u ,b u ,v f ,f v ;
1 if e∈∈{{bnnv11,unna11,fnnu11,vnne11,fnnw11}.n 1 n 1 n 1 n−1 1 n 2 n−1 1 n 2}
Thus e (1)=15n and e (0)=15n.
f f
Case 2 n is odd.
Define a vertex labeling f :V(G) 0,1 as follows:
→{ }
0 if x=t;
0 if x v ,a ,d ,f ,g , i 1,2,...,n 1,n ;
i i i i i
f(x)=11 iiff xx∈∈={{wui,,bii,ci,1e,i3,,h.i.}}.,,ni∈∈{{21,,n2,;...,n−−1,n}};
i
∈{ − }
11n+10 if x=wi,1i1n∈+{21,4,...,n−3,n−1}.
Thusvf(1)= andvf(0)= . Theinducededgelabelingf∗ :E(G) 0,1
2 2 →{ }
is f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e tu ,u v ,a b ,e f ,c d ,g h ,tc ,tb ,g u , i 1,2,...,n 1,n ;
i i i i i i i i i i i i i i i
∈{ } ∈{ − }
0 if e ta ,a v ,e u , i 1,2,...,n 1,n ;
i i i i i
f∗(e)=01010 iiiiifffff eeeee∈∈∈∈∈∈{{{{{{ubbddiiiiivuwwuiii+ii++,,12vv1,,,iiuvwwviiiiiafv}}iii+,,++211ii}},,∈∈f,diii{{uv∈12i∈i++,,{3411{,,,,1v..v,..ii2ed..,i,,i+.+nn.11.−−},,,cni23−iu,,−i∈nn+3}−1{,;,1nu1},}i2−c;,i.+2.}1.;,,gniv−i+21,,nvi−gi+11}};,1≤i≤n−1;
0 if e=e w , i 1,3,...,n 2,n ;
1100 iiiiffff eeee∈∈=∈{{{febuiiwninwviui+1i1,,1,fu,vi∈nnwiav{i∈11+,,1{fd}1nn,,uv3i11,,,∈.vv.nn{.ed2,−11n,},4−.c,n.4u..,}1,n,nu−n−c213},,;gnn−v1,1v}n;g1,bn−1u1,bnu2,vn−1f1,fnv2,fnw1};
Thus e (1)=15n and e (0)=15n.
f f
Therefore f satisfies the conditions v (1) v (0) 1 and e (1) e (0) 1. So, f
f f f f
| − | ≤ | − | ≤
admits cordial labeling on G. Hence G is cordial.
2
98 UMPrajapatiandRMGajjar
Theorem 2.7 The graph obtained by duplicating all the vertices of the armed helm AH is
n
cordial.
Proof Let V(AH ) = t u ,v ,w /1 i n and E(AH ) = tu ,u v ,v w /1
n i i i n i i i i i
{ } { ≤ ≤ } { ≤
i n u u ; 1 i n 1 u u . Let G be the graph obtained by duplicating all
≤ } { i i+1 ≤ ≤ − S} { n 1}
the vertices in AH . Let t,u ,u , ,u ,v ,v , ,v ,w ,w , ,w be the new vertices
S n ′ ′1 ′2S··· ′n 1′ 2′ ··· n′ 1′ 2′ ··· n′
of G by duplicating t,u ,u , ,u ,v ,v , ,v ,w ,w , ,w respectively. Then V(G) =
1 2 n 1 2 n 1 2 n
··· ··· ···
t,t u ,v ,w ,u ,v ,w /1 i n andE(G)= tu ,u v ,v w ,w v ,w v ,u v ,tu ,u v ,tu ;
{ ′} { i i i ′i i′ i′ ≤ ≤ } { i i i i i i′ i i i′ i i′ ′ i ′i i ′i
1 i n u u ,u u ,u u ; 1 i n 1 u u ,u u ,u u . Therefore V(G) =
≤ ≤S } { i i+1 ′i i+1 i ′i+1 ≤ ≤ − } { n 1 ′n 1 n ′1} | |
6n+2 and E(G) =12n. Define a vertex labeling f :V(G) 0,1 as follows:
S| | S →{ }
1 if x=t;
f(x)=01 iiff xx∈∈{{uvii,,wu′ii,,wvi′i′}},, ii∈∈{{11,,22,,......,,nn−−11,,nn}};;
0 if x=t′.

Thus v (1)=3n+1 and v (0)=3n+1. The induced edge labeling f :E(G) 0,1 is
f f ∗
→{ }
f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e tu ,u v ,u v ,v w ,u v ,tu , i 1,2,...,n 1,n ;
∈{ i i i i i′ i i′ ′i i ′i} ∈{ − }
0 if e v w ,tu ,w v , i 1,2,...,n 1,n ;
f∗(e)=0 if e∈∈{{uiiuii+1′,ui′iuii+1i′,}uiu′i∈+1{}, i∈{1,2−,...,n}−2,n−1};
0 if e u u ,u u ,u u .
 ∈{ n 1 ′n 1 n ′1}
Thus e (1)=6n and e (0)=6n. Therefore, f satisfies the conditions v (1) v (0) 1 and
f f f f
| − |≤
e (1) e (0) 1. So, f admits cordial labeling on G. Hence G is cordial.
f f
| − |≤ 2
Theorem 2.8 The graph obtained by duplicating all the vertices other than the rim vertices of
the armed helm AH is cordial.
n
Proof Let V(AH ) = t u ,v ,w /1 i n and E(AH ) = tu ,u v ,v w /1
n i i i n i i i i i
{ } { ≤ ≤ } { ≤
i n u u u u /1 i n 1 . Let G be the graph obtained by duplicating all
≤ }∪{ n 1} { i i+1 ≤S≤ − }
the vertices other than the rim vertices in AH . Let t,v ,v , ,v ,w ,w , ,w be the
S n ′ 1′ 2′ ··· n′ 1′ 2′ ··· n′
new vertices of G by duplicating t,v ,v , ,v ,w ,w , ,w respectively. Then V(G) =
1 2 n 1 2 n
··· ···
t,t u ,v ,w ,v ,w /1 i n and E(G) = tu ,u v ,v w ,w v ,w v ,u v ,tu /1 i
{ ′} { i i i i′ i′ ≤ ≤ } { i i i i i i′ i i i′ i i′ ′ i ≤ ≤
n u u /1 i n 1 u u . Therefore V(G) = 5n+2 and E(G) = 8n. Using
} {Si i+1 ≤ ≤ − }∪{ n 1} | | | |
parity of n, we have the following cases:
S
Case 1 n is even.
Define a vertex labeling f :V(G) 0,1 as follows:
→{ }
0 if x=t;
f(x)=

0 if x∈{ui,wi′}, i∈{1,2,...,n−1,n}

CordialityintheContextofDuplicationinWebandArmedHelm 99
and
1 if x v ,w , i 1,2,...,n 1,n ;
i i
∈{ } ∈{ − }
1 if x=t′;
f(x)=1 if x=vi′, i∈{1,3,...,n−3,n−1};
0 if x=v , i 2,4,...,n 2,n .
5n+2  5ni′+2∈{ − }
Thus v (1)= and v (0)= . The induced edge labeling f :E(G) 0,1
f f ∗
2 2 →{ }
is f (uv)= f(u) f(v), for every edge e=uv E.
∗
| − | ∈
Therefore
1 if e u v ,v w ,t u , i 1,2,...,n 1,n ;
∈{ i i i i′ ′i i} ∈{ − }
1 if e=u v , i 1,3,...,n 3,n 1 ;
f∗(e)=100 iiifff eee=∈={wutiiiuuvii′ii′,+,v1ii,w∈∈ii}{{∈,2{,i41∈,,.2{.,1..,,.2n.,,−−.n..−2,,nn2,}−−n;1−,}n1}};;
0 if e=u v , i 2,4,...,n 2,n ;
Thus ef(1)=4n and ef(0)00=4iiffn.ee=∈{wuiinvi′ui′,1}i.∈∈{{1,3,...,n−−3,n}−1};
Case 2 n is odd.
Define a vertex labeling f :V(G) 0,1 as follows:
→{ }
1 if x=t;
′
1 if x v ,w , i 1,2,...,n 1,n ;
i i
f(x)=10 iiff xx∈=={vt;i′, i∈}{1,∈3,{...,n−2,n−}; }
0 if x u ,w , i 1,2,...,n 1,n ;
5n+3 0 if x∈=5{vni′,+i i1∈i′}{2,4∈,.{..,n−3,n−−1}.}
Thus v (1)= and v (0)= . The induced edge labeling f :E(G) 0,1
f f ∗
2 2 →{ }
is f (uv)= f(u) f(v), for every edge e=uv E. Therefore
∗
| − | ∈
1 if e u v ,v w ,t u , i 1,2,...,n 1,n ;
∈{ i i i i′ ′i i} ∈{ − }
1 if e=u v , i 1,3,...,n 2,n ;
f∗(e)=10 iiff ee=wtiiuvi′i′,,viw∈∈{{,2,i4,..1.,,2n,−−...3,,nn}−11,}n; ;
i i i
∈{ } ∈{ − }
0 if e=uiui+1, i∈{1,2,...,n−2,n−1}