cordiality of transformation graphs of cycles · 2016-07-06 · k. manjula bangalore institute of...
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International Journal of Mathematics Research.ISSN 0976-5840 Volume 8, Number 2 (2016), pp. 113- 125© International Research Publication Househttp://www.irphouse.com
Cordiality of Transformation Graphs of Cycles
Chandrakala S.B.Nitte Meenakshi Institute of Technology,
Bangalore, India.
K. ManjulaBangalore Institute of Technology,
Bangalore, India.
Abstract
A graph is said to be cordial if it has a 0-1 labeling that satisfies certain properties.In this paper we show that transformation graphs of cycle are cordial. We also showthat total graph of sunlet graph, comb and star are cordial.
AMS subject classification: 05C78.Keywords: Transformation Graph, Cordial Graph.
1. Introduction
All graphs G considered here are finite, undirected and simple. We refer to [1] forunexplained terminology and notations. In 2001, Wu and Meng [3] introduced somenew graphical transformations which generalizes the concept of the total graph. As isthe case with the total graph, these generalizations referred to as transformation graphsGxyz have V (G) ∪ E(G) as the vertex set. The adjacency of two of its vertices isdetermined by adjacency and incidence nature of the corresponding elements in G.
Let α, β be two elements of V (G) ∪ E(G). Then associativity of α and β is takenas + if they are adjacent or incident in G, otherwise −. Let xyz be a 3-permutation ofthe set {+, −}. The pair α and β is said to correspond to x or y or z of xyz if α andβ are both in V (G) or both are in E(G), or one is in V (G) and the other is in E(G)
respectively. Thus the transformation graph Gxyz of G is the graph whose vertex set isV (G) ∪ E(G). Two of its vertices α and β are adjacent if and only if their associativityin G is consistent with the corresponding element of xyz.
In particular the transformation graph G++− of G is the graph with vertex set V (G)∪E(G) in which the vertices u and v are joined by an edge if one of the following holds
114 Chandrakala S.B. and K. Manjula
1. both u, v ∈ V (G) and u and v are adjacent in G
2. both u, v ∈ E(G) and u and v are adjacent in G
3. one is in V (G) and the other is in E(G) and they are not incident with each otherin G.
The transformation graphs are investigated in [4], [5] and [6].A graph labeling is an assignment of integers to the vertices or edges, or both,
subject to certain conditions. A mapping f : V (G) → {0, 1} is called a binary labelingof the graph G. For each v ∈ V (G), f (v) is called the vertex label of the vertex v
under f and for an edge uv the induced edge labeling g : E(G) → {0, 1} is given byg(uv) = |f (u) − f (v)|. Then f is called a cordial labeling of G if the number ofvertices labeled 0 and the number of vertices labeled 1 differs by at most 1, and, thenumber of edges labeled 0 and the number of edges labeled 1 differs by at most 1. Agraph G is cordial if it admits a cordial labeling. This concept is introduced by Cahit[2]. For more cordial graphs we refer to [8], [9], [10], [12] and [11].
For convenience, the transformation graph Gxyz is partitioned into Gxyz = Sx(G) ∪Sy(G)∪Sz(G) where Sx(G), Sy(G) and Sz(G) are the edge-induced subgraphs of Gxyz.The edge set of each of which is respectively determined by x, y and z of the permutationxyz. Sx(G) ∼= G when x is + and Sx(G) ∼= G when x is −. Sy(G) ∼= L(G) when y is+ and Sy(G) ∼= L(G) when y is −. When z is +, α, β ∈ V (Gxyz) are adjacent in Sz(G)
if they are incident with each other in G. When z is −, α, β are adjacent in Sz(G) if theyare not incident in G.
The following notations are used in relation to labeling of Gxyz:Let V0 and V1 denote the set of vertices of Gxyz labeled 0 and 1.E0 and E1 denote set of edges of Gxyz labeled 0 and 1 respectively.E0(Sx) and E1(Sx) denote set of edges labeled 0 and 1 in Sx(G). Similar meanings
are associates with E0(Sy), E1(Sy), E0(Sz) and E1(Sz) .For P xyz
n xyz ∈ {+ + −, + − −, − + +, − + −, − − +, − − −, + + +} wedefine a vertex labeling f : V (P xyz
n ) → {0, 1} and specify the induced edge labelingg : E(P xyz
n ) → {0, 1} then show that f is a cordial labeling.
2. Cordiality of Transformation Graphs of Cycle
Let Cn : v1 − v2 − v3 − . . . − vn(n ≥ 3) be the cycle on n vertices and ei = vivi+1(1 ≤i ≤ n − 1) and en = vnv1 be the edges of Cn.
Theorem 2.1. For any positive integer n ≥ 3
(a) C++−n is cordial
(b) C+++n is cordial
(c) C−++n is cordial when n ≡ 0, 2, 3, 5, 6, 7(mod 8)
Cordiality of Transformation Graphs of Cycles 115
(d) C+−−n is cordial when n ≡ 0, 1, 2, 3, 5, 6(mod 8)
(e) C+−+n is cordial when n ≡ 0, 2, 3, 5, 6, 7(mod 8)
(f) C−−−n is Cordial when n ≡ 0, 1, 3, 4, 5, 7(mod 8)
(g) C−+−n is Cordial when n ≡ 0, 1, 2, 3, 5, 6(mod 8)
(h) C−−+n is Cordial when n ≡ 0, 1(mod 4).
Proof.
(a) For n ≥ 3 the vertices of C−+−n are labeled as in table 1:
n )( ivf )( ief 10 VV −
Even
−≤≤
otherwise
ni
,0
12
1,1
+≤≤
otherwise
ni
,0
12
1,1
0
Odd
≤≤
otherwise
ni
,02
1,1
≤≤
otherwise
ni
,02
1,1
Table 1: Cordial labeling of C++−n .
|E0| and |E1| are calculated from the labeling of the vertices of V (C++−n ), and are
given in the table 2:
n )(0 xSE )(0 ySE )(0 zSE � )(1 xSE )(1 ySE )(1 zSE 10 ~ EE
Even
22
−n
22
−n
4
)2(2
−+−
nnn
4
)4)(2( −++
nn
2 2
4
)2(2
−+−
nnn
4
)4)(2( −++
nn 0
Odd 2
742+− nn
2
72−n
1
Table 2: |E0| and |E1| of C++−n .
Therefore C++−n is cordial.
(b) The vertices of C+++n are labeled as below:
f (vi) = 1 for 1 ≤ i ≤ n
f (ei) = 1 for 1 ≤ i ≤ n
116 Chandrakala S.B. and K. Manjula
Here |V0| − |V1| = 0|E0| and |E1| are calculated from the labeling of the vertices of C+++
n , and are:
|E0| = |E0(Sx)| + |E0(Sy)| + |E0(Sz)| = n + n + 0 = 2n
|E1| = |E1(Sx)| + |E1(Sy)| + |E1(Sz)| = 0 + 0 + 2n = 2n
Therefore |E0| − |E1| = 0.
Therefore C+++n is cordial.
(c) When 3 ≤ n ≤ 7, the vertices of C−++n are labeled as in Table 3 which admits
cordial labeling.
n f (v1)f (v2)...f (vn) f (e1)f (e2)...f (en)
3 101 1004 1010 11005 10101 110006 101010 1100107 1010101 1100100
Table 3: Cordial labeling of C−++n for the case n < 7.
when n ≥ 8, the vertex labeling of C−++n are given in Table 4 and |E0| and |E1|
of C−++n are counted in Table 5.
n )( ivf )( ief 10 ~ VV
rn 8=
28 += rn 1
≤≡≤+
≤≡≤
otherwise
nir
ri
1)2(mod0240
4)4(mod3,030
0 38 += rn
58 += rn78 += rn
�
0
=
−≤≡≤+
+≤≡≤
otherwise
ni
nir
ri
10
1)2(mod0640
44)4(mod3,030
�
68 += rn1
≤≡≤+
+≤≡≤
otherwise
nir
ri
1)2(mod0640
44)4(mod3,030
Table 4: cordial labeling of C−++n for the case n ≥ 8.
Therefore C−++n is cordial.
(d) For n = 3, 5and6, the vertices of C+−−n are labeled as in Table 6 which admits
cordial labeling.
For n ≥ 8, the vertices of C+−−n are labeled as in Table 7. In all the cases
|V0| − |V1| = 0.
Cordiality of Transformation Graphs of Cycles 117
n )(0 xSE )(0 ySE )(0 zSE )(1 xSE )(1 ySE )(1 zSE 10 ~ EE
�
rn 8=
∑
−
=
2
2
1
2
n
i
i r2 n2
42
2
4
1
−+∑
−
=
ni
n
i
)2( rn − n
0
28 += rn1
38 += rn
2
32
2
3
1
−+∑
−
=
ni
n
i
32 +r 1−n ∑
−
=
2
3
1
2
n
i
i32 −− rn
1+n
1
58 += rn0
78 += rn1
68 += rn
∑
−
=
2
2
1
2
n
i
i 22 +r n2
42
2
4
1
−+∑
−
=
ni
n
i
22 −− rnn 1
Table 5: |E0| and |E1| of C−++n .
n f (v1)f (v2)...f (vn) f (e1)f (e2)...f (en)
3 110 0015 11000 101016 110010 101010
Table 6: Cordial labeling of C+−−n for the case n < 8.
|E0| and |E1| of C+−−n for the case n ≥ 8 are given Table 8.
From the Table 8 it is evident that C+−−n .
(e) For 3 ≤ n ≤ 8, the vertices of C+−+n are labeled as in Table 9 which admits cordial
labeling.
For n ≥ 8, f is defined as in (d) which admits cordial labeling.
(f) For n = 3, 4, 5, 7, the vertices of C−−−n are labeled as in Table 10 which admits
cordial labeling.
For n ≥ 8, the vertices of C−−−n are labeled as in Table 11 and |E0| and |E1| of
C−−−n for the case n ≥ 8 are given in Table 12.
Therefore C−−−n is cordial.
(g) For 3 ≤ n ≤ 8, the vertices of C−+−n are labeled as in Table 13 which admits cordial
labeling.
For n ≥ 8, the vertices of C−+−n labeled as in Table 14 and |E0| and |E1| of C−+−
n
for the case n ≥ 8 are given table 15:
Therefore C−+−n is cordial.
118 Chandrakala S.B. and K. Manjula
n )( ief )( ivf10 ~ VV
rn 8=
28 += rn 0
≤≡≤+
≤≡≤
otherwisenir
ri
1)4(mod2,0240
4)4(mod3,030�
0 ,18 += rn,38 += rn
58 += rn�����
1
=
−≤≡≤+
+≤≡≤
otherwiseni
nir
ri
10
1)4(mod2,0640
44)4(mod3,030
�68 += rn0
Table 7: Cordial labeling of C+−−n for the case n ≥ 8.
n )(0 xSE
)(0 ySE )(0 zSE )(1 xSE )(1 ySE )(1 zSE 10 ~ EE
�
rn 8=
r2 ∑
−
=
2
2
1
2
n
i
i
( ) +− rn 2
)2(22
rnn
−
− rn 2− 2
42
2
4
1
−+∑
−
=
ni
n
i
( ) +− rn 2
)2(22
rnn
−
−
0
28 += rn1
38 += rn
32 +r 2
32
2
3
1
−+∑
−
=
ni
n
i 2
322−− nn 32 −− rn
∑
−
=
2
3
1
2
n
i
i 2
322+− nn
1
58 += rn0
78 += rn1
68 += rn22 +r ∑
−
=
2
2
1
2
n
i
i 2
)2( −nn 22 −− rn2
42
2
4
1
−+∑
−
=
ni
n
i 2
)2( −nn1
Table 8: |E0| and |E1| of C+−−n for the case n ≥ 8
(h) Vertices of V (C−−+n ) are as in Table 16 and |E0| and |E1| of C−−+
n for the casen ≥ 8 are given table 17:
�
3. Cordiality of G+++ when G isomorphic to Sunlet,Comb and Star Graphs
We recall the following definitions:
Definition 3.1. Corona G1 �G2 of G1 and G2 is the graph obtained by taking one copyof G1(which has n1 vertices) and n1 copies of G2 and then joining ith vertex of G1 toevery vertex in the ith copy of G2. So the order G1 � G2 is n1 + n1n2 where n2 is theorder of G2 and its size is m1 + n1n2 + n1m2 where mi is the size of Gi , ∀ i = 1, 2.
Cordiality of Transformation Graphs of Cycles 119
n f (v1)f (v2) · · · f (vn) f (e1)f (e2) · · · f (en)
3 001 0114 1101 10005 11001 010106 110010 0101017 1100101 0101010
Table 9: Cordial labeling ofC+−+n for the case 3 ≤ n ≤ 8.
n f (v1)f (v2) · · · f (vn) f (e1)f (e2) · · · f (en)
3 010 1104 0101 11005 01010 110017 0101010 1100101
Table 10: Cordial labeling ofC−−−n for the case n < 8.
Definition 3.2. Pn � K1 (n ≥ 3) is called comb graph of order 2n.Definition 3.3. Cn � K1 (n ≥ 3) is called sunlet graph of order 2n.
Theorem 3.4. For any positive integer n ≥ 3 total graph of sunlet graph Sn is cordial.
Proof. Let vi(1 ≤ i ≤ n) be the consecutive vertices on the cycle Cn, v′i be the pendant
vertices adjacent to vi respectively. Let ei = vivi+1 be the edges on the cycle and e′i be
the pendant edges incident to vi respectively.Define a binary labeling f : V (S+++
n ) → {0, 1} as in Table 18 and |E0| and |E1| ofS+++
n for the case n ≥ 6 are given Table 19:Therefore S+++
n is cordial. �
n )( ivf )( ief 10 ~ VV
)8(mod4,0≡n �
evenisi
oddisi
1
0
≡
otherwise
i
0
)4(mod2,11
0
)8(mod5,1≡n
)8(mod7≡n
−=
−=
−≤≡
−≤≡
10
,21
3)4(mod3,00
5)4(mod2,11
ni
nni
ni
ni
�
)8(mod3≡n
=
>
evenisii
oddisandi
111
10�
≡
otherwise
i
0
)4(mod4,31
Table 11: cordial labeling of C−−−n for the case n ≥ 8.
120 Chandrakala S.B. and K. Manjula
n )(0 xSE )(0 ySE )(0 zSE )(1 xSE )(1 ySE )(1 zSE 10 ~ EE
�
)8(mod
4,0≡n
∑
−
=
2
2
1
2
n
i
i ∑
−
=
4
4
1
8
n
i
i 2
)2( −nn2
42
2
4
1
−+∑
−
=
ni
n
i ∑
−
=
2
2
1
2
n
i
i 2
)2( −nn0
)8(mod
5,1≡n
2
32
2
3
1
−+∑
−
=
ni
n
i
122
3
1
−∑
−
=
n
i
i
2
)1( 2−n
∑
−
=
2
3
1
2
n
i
i
2
12
2
3
1
−+∑
−
=
ni
n
i
2
122−− nn
1
)8(mod
7≡n
∑
−
=
2
3
1
2
n
i
i 2
32
2
3
1
−+∑
−
=
ni
n
i
1
)8(mod
3≡n 1
Table 12: |E0| and |E1| of C−−−n for the case n ≥ 8
n f (v1)f (v2) · · · f (vn) f (e1)f (e2) · · · f (en)
3 101 1004 11100 00015 10101 000116 10101 1101017 1010101 1101000
Table 13: Cordial labeling ofC−+−n for the case 3 ≤ n ≤ 8.
Theorem 3.5. Total graph of Comb graph is cordial.
Proof. Let G = Pn
⊙K1(n ≥ 3) be a comb on 2n vertices. Let vi(1 ≤ i ≤ n) be the
consecutive vertices of degree 2, v′i(1 ≤ i ≤ n) be the pendant vertices adjacent to vi
respectively. Let ei = vivi+1(1 ≤ i ≤ n − 1) be the edges and e′i(1 ≤ i ≤ n) be the
pendant edges incident to vi respectively. Define a binary labeling f : V (G) → {0, 1}as in Table 20 and |E0| and |E1| of total graph of comb for the case n ≥ 6 are as shownin Table 21.
�
n )( ivf )( ief 10 ~ VV
n=8r,
8r+2
evenisi
oddisi
0
1
≤≡≤+
≤≡
otherwise
nir
ri
1)4(mod4,2240
4)4(mod0,30
0 n=8r+6
n=8r+1
8r+3,
8r+5
≤≡≤+
+≤≡
otherwise
nir
ri
1)4(mod4,2640
44)4(mod0,30
Table 14: cordial labeling of C−+−n for the case n ≥ 8.
Cordiality of Transformation Graphs of Cycles 121
n )(0 xSE )(0 ySE )(0 zSE )(1 xSE )(1 ySE )(1 zSE 10 ~ EE
)8(mod0≡n
∑
−
=
2
2
1
2
n
i
i r22
)2( −nn
2
42
2
4
1
−+∑
−
=
ni
n
i
rn 2− 2
)2( −nn 0
)8(mod2≡n1
)8(mod1≡n
2
32
2
3
1
−+∑
−
=
ni
n
i
12 +r2
)1( 2−n
∑
−
=
2
3
1
2
n
i
irn 21−−
2
122−− nn
1
)8(mod3≡n0
)8(mod5≡n1
)8(mod6≡n
∑
−
=
2
2
1
2
n
i
i 22 +r2
)2( −nn
2
42
2
4
1
−+∑
−
=
ni
n
i
rn 22 −−
2
)2( −nn
1
Table 15: |E0| and |E1| of C−+−n for the case n ≥ 8
n )( ivf )( ief 10 VV −
)4(mod0≡n
≡
otherwise
i
1
)4(mod3,00
0 )4(mod1≡n
≡
otherwise
i
1
)4(mod3,00
=
−≤≡≤
otherwise
ni
ni
1
,3,20
2)4(mod3,060
Table 16: Cordial labeling of C−−+n .
Theorem 3.6. Total graph of Star K1,n wheren ≡ 0, 1, 3, 4, 5, 6(mod 8) is cordial.
Proof. Let v0 be the central vertex and vi(1 ≤ i ≤ n) be the pendant vertices of K1,n.
Let ei = v0vi(1 ≤ i ≤ n) be the pendant edges.Define a binary labeling f : V (G) → {0, 1} as in Table 22 and |E0| and |E1| of total
graph of Star for the case n ≥ 6 are as shown in Table 23. �
4. Conclusion
We did not get the cordial labeling of Gxyz for some specific values of n when G isisomorphic to path, cycle and star. One of the future work on this paper is to find somemore standard graphs G for which the transformation graphs are cordial.
122 Chandrakala S.B. and K. Manjula
n )(0 xSE )(0 ySE )(0 zSE )(1 xSE )(1 ySE )(1 zSE 10 EE −
�
)4(mod0≡n
∑
−
=
4
4
1
8
n
i
i ∑
−
=
4
4
1
8
n
i
i 2
nn + ∑
−
=
2
2
1
2
n
i
i ∑
−
=
2
2
1
2
n
i
i 2
n0
)4(mod1≡n ∑
−
=
+−
2
5
1
2)4(
n
i
in
+− rn 2
∑=
−
r
i
i1
)12(42
1−n ∑
−
=
+−
2
3
1
22
1n
i
in
122
3 2
3
1
−+−
∑
−
=
n
i
in 2
2
1+
−n
0
Table 17: |E0| and |E1| of C−−+n
)6( ≥n )( ivf )( ivf ′ )( ief )( ief ′10 VV −
rn 4=
0
)1( ni ≤≤
1
)1( ni ≤≤
≤≤
otherwise
ri
,0
1,1
≤≤
otherwise
ri
,1
1,00 14 += rn
34 += rn +≤≤
otherwise
ri
,0
11,1
+≤≤
otherwise
ri
,1
11,0
Table 18: Cordial labeling of S+++n for the case n ≥ 6.
)6( ≥n )( 10 SE )( 20 SE )( 30 SE )( 11 SE )( 21 SE )( 31 SE 10 EE − �
n=4r, 4r+1 n n 3n-2r n 2n n+2r 0, 1
34 += rn n n 3n-2r-2 n 2n n+2r+2 -1
Table 19: |E0| and |E1| of S+++n
for the case n ≥ 6
Cordiality of Transformation Graphs of Cycles 123
)6( ≥n )( ivf )( ivf ′ )( ief )( ief ′ 10 VV −
rn 4=
0
)1( ni ≤≤
1
)1( ni ≤≤
−≤≤−
otherwise
nir
1
210
−≤≤−
otherwise
nir
0
111
0 14 += rn
−=
≤≤
otherwiseni
ri
011
21
=
≤≤
otherwiseni
ri
10
10
24 += rn
−=
−≤≤
otherwiseni
ri
011
111
=
−≤≤
otherwiseni
ri
10
110
34 += rn
Table 20: Cordial labeling of total graph of Comb for the case n ≥ 6.
)6( ≥n )(0 xSE )(0 ySE)(0 zSE
)(1 xSE )(1 ySE )(1 zSE 10 ~ EE
�
rn 4=
n-1
n-2 3n-2r
n
2n-2 n+2r-2 1
14 += rn n-1
3n-2r-2
2n-3
n+2r
0
24 += rnn-2 2n-2
1
34 += rn 0
Table 21: |E0| and |E1| of total graph of comb for the case n ≥ 6.
)6( ≥n )( ivf )( ief 10 ~ VV
rn 8=
≤≤
otherwise
ni
,0
21,1
≤≤++
−
≤≤
otherwise
nirn
andrn
i
,0
12
21,1
1
18 += rn1
28 += rn1
38 += rn1
48 += rn1
58 += rn1
68 += rn
≤≤++
−−
≤≤
otherwise
nirn
andrn
i
,0
22
12
1,1
1
Table 22: Cordial labeling of total graph of Star for the case n ≥ 6.
124 Chandrakala S.B. and K. Manjula
)6( ≥n )(0 xSE )(0 ySE )(0 zSE )(1 xSE )(1 ySE )( 31 SE 10 ~ EE
�
rn 8=
2
n
×
2
2
2
2
nnrn
n22
2−+
2
n
×
22
nn
rn
22
+
0
18 += rn 1
28 += rn 1
38 += rn 0
48 += rn 0
58 += rn 1
68 += rn 222
3−− r
n 222
++ rn 1
Table 23: |E0| and |E1| of total graph of Star for the case n ≥ 6.
Acknowledgements
The authors wish to express their gratitude to Dr. B. Sooryanarayana for his helpfulcomments and suggestions.
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