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TOTAL MEAN CORDIAL LABELING OF SOME CYCLE RELATED GRAPHS
TOTAL MEAN CORDIAL LABELING OF SOME CYCLE RELATED GRAPHS
Journal of Applied Mathematics & Informatics. 2015. Jan, 33(1_2): 101-110
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : July 17, 2014
  • Accepted : October 10, 2014
  • Published : January 30, 2015
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About the Authors
R. PONRAJ
S. SATHISH NARAYANAN

Abstract
A Total Mean Cordial labeling of a graph G = ( V,E ) is a function f : V ( G ) → {0, 1, 2} such that where x, y V ( G ), xy E ( G ), and the total number of 0, 1 and 2 are balanced. That is | evf ( i ) − evf ( j )| ≤ 1, i, j ∈ {0, 1, 2} where evf ( x ) denotes the total number of vertices and edges labeled with x ( x = 0, 1, 2). If there is a total mean cordial labeling on a graph G , then we will call G is Total Mean Cordial. Here, We investigate the Total Mean Cordial labeling behaviour of prism, gear, helms. AMS Mathematics Subject Classification : 05C78.
Keywords
1. Introduction
Terminology and notations in graph theory we refer Harary [2] . New terms and notations shall, however, be specifically defined whenever necessary. By a graph G = ( V,E ) we mean a finite, undirected graph with neither loops nor mul-tiple edges. The product graph G 1 × G 2 is defined as follows: Consider any two points u = ( u 1 , u 2 ) and v = ( v 1 , v 2 ) in V = V 1 × V 2 . Then u and v are adjacent in G 1 × G 2 whenever [ u 1 = v 1 and u 2 adj v 2 ] or [ u 2 = v 2 and u 1 adj v 1 ]. The join of two graphs G 1 and G 2 is denoted by G 1 + G 2 and whose vertex set is V ( G 1 + G 2 ) = V ( G 1 ) ∪ V ( G 2 ) and edge set E ( G 1 + G 2 ) = E ( G 1 ) ∪ E ( G 2 ) ∪ { uv : u V ( G 1 ), v V ( G 2 )}. The order and size of G are denoted by p and q respectively. Ponraj, Ramasamy and Sathish Narayanan [3] introduced the concept of Total Mean Cordial labeling of graphs and studied about their be-havior on Path, Cycle, Wheel and some more standard graphs. In [4] , Ponraj and Sathish Narayanan proved that
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is Total Mean Cordial if and only if n = 1 or 2 or 4 or 6 or 8. Also in [5] , Ponraj, Ramasamy and Sathish Narayanan studied about the Total Mean Cordiality of Lotus inside a circle, bistar, flower graph, K 2, n , Olive tree,
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In this paper, we investigate the Total Mean Cordiality of some cycle related graphs. Let x be any real number. Then the symbol ⌈ x ⌉ stands for the smallest integer greater than or equal to x .
2. Main results
Definition 2.1. A Total Mean Cordial labeling of a graph G = ( V,E ) is a function f : V ( G ) → {0, 1, 2} such that
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where x, y V ( G ), xy E ( G ), and the total number of 0, 1 and 2 are balanced. That is | evf ( i ) - evf ( j )| ≤ 1, i, j ∈ {0, 1, 2} where evf ( x ) denotes the total number of vertices and edges labeled with x ( x = 0, 1, 2). If there exists a total mean cordial labeling on a graph G , we will call G is Total Mean Cordial.
Prisms are graphs of the form Cm × Pn . We now look into the graph prism Cn × P 2 .
Theorem 2.2. Prisms are Total Mean Cordial.
Proof . It is clear that p + q = 5 n . Let V ( Cn × P 2 ) = { ui , vi : 1 ≤ i n } and E ( Cn × P 2 ) = { u 1 un , v 1 vn }∪{ uivi : 1 ≤ i n }∪{ ui u i +1 , vi v i +1 : 1 ≤ i n −1}.
Case 1. n ≡ 0 (mod 6).
Let n = 6 t and t > 0. Define a map f : V ( Cn × P 2 ) → {0, 1, 2} by
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f ( u 2 t +1 ) = 0, f ( v 2 t +1 ) = f ( v 6 t ) = 2, f ( u 6 t ) = 1. In this case evf (0) = evf (1) = evf (2) = 10 t .
Case 2. n ≡ 1 (mod 6).
Let n = 6 t + 1 and t > 0. Define a map f : V ( Cn × P 2 ) → {0, 1, 2} by
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f ( v 2 t +1 ) = f ( v 2 t +2 ) = 1. Here evf (0) = evf (2) = 10 t + 2, evf (2) = 10 t + 1.
Case 3. n ≡ 2 (mod 6).
Let n = 6 t + 2 and t > 0. Define a map f : V ( Cn × P 2 ) → {0, 1, 2} by
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f ( u 6 t +2 ) = 1, f ( v 6 t +2 ) = 2. In this case evf (0) = evf (2) = 10 t + 3, evf (1) = 10 t + 4.
Case 4. n ≡ 3 (mod 6).
Let n = 6 t + 3 and t ≥ 0. A Total Mean Cordial labeling of C 3 × P 2 is given in figure 1 .
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Assume t > 0. Define a map f : V ( Cn × P 2 ) → {0, 1, 2} by
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f ( u 2 t +2 ) = 0, f ( u 6 t +3 ) = 1, f ( v 2 t +2 ) = f ( v 6 t +3 ) = 2. In this case evf (0) = evf (1) = evf (2) = 10 t + 5.
Case 5. n ≡ 4 (mod 6).
Let n = 6 t + 4 and t ≥ 0. A Total Mean Cordial labeling of C 4 × P 2 is given in figure 2 .
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Assume t > 0. Define a map f : V ( Cn × P 2 ) → {0, 1, 2} by
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In this case evf (0) = evf (1) = 10 t + 7, evf (2) = 10 t + 6.
Case 6. n ≡ 5 (mod 6).
Let n = 6 t - 1 and t > 0. Define a map f : V ( Cn × P 2 ) → {0, 1, 2} by
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f ( u 6 t -1 ) = 1, f ( v 6 t -1 ) = 2. In this case evf (0) = evf (2) = 10 t - 2, evf (1) = 10 t - 1. □
The gear graph Gn is obtained from the wheel Wn = Cn + K 1 where Cn is the cycle u 1 u 2 . . . unu 1 and V ( K 1 ) = { u } by adding a vertex between every pair of adjacent vertices of the cycle Cn .
Theorem 2.3. The gear graph Gn is Total Mean cordial.
Proof . Let V ( Gn ) = V ( Wn ) ∪ { vi : 1 ≤ i n } and E ( Gn ) = E ( Wn ) ∪ { uivi , vj u j +1 : 1 ≤ i n , 1 ≤ j n } − E ( Cn ). Clearly p + q = 5 n + 1.
Case 1. n ≡ 0 (mod 12).
Let n = 12 t and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t +1 ) = 0, f ( v 12 t ) = 1.
Case 2. n ≡ 1 (mod 12).
Let n = 12 t + 1 and t > 0. Assign the label to the vertices ui (1 ≤ i ≤ 12 t ), vi (1 ≤ i ≤ 12 t −1) as in case 1. Then put the labels 0, 1, 2 to the vertices v 12 t , u 12 t +1 , v 12 t +1 , respectively.
Case 3. n ≡ 2 (mod 12).
Let n = 12 t + 2 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 9 t +2 ) = f ( v 12 t +2 ) = 2, f ( u 12 t +2 ) = 0, f ( u 12 t +1 ) = 1.
Case 4. n ≡ 3 (mod 12).
Let n = 12 t - 9 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t -3 ) = 0, f ( v 12 t -9 ) = 1.
Case 5. n ≡ 4 (mod 12).
Let n = 12 t − 8 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 12 t -8 ) = 1, f ( v 12 t -8 ) = 2.
Case 6. n ≡ 5 (mod 12).
Let n = 12 t − 7 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t -2 ) = 0, f ( v 5 t -2 ) = 1, f ( u 9 t -4 ) = 2, f ( v 12 t -7 ) = 1,
Case 7. n ≡ 6 (mod 12).
Let n = 12 t − 6 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 9 t -4 ) = 2, f ( v 12 t -6 ) = 2.
Case 8. n ≡ 7 (mod 12).
Let n = 12 t − 5 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 12 t -6 ) = f ( u 12 t -5 ) = 1, f ( v 12 t -6 ) = 0, f ( v 12 t -5 ) = 2.
Case 9. n ≡ 8 (mod 12
Let n = 12 t − 4 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t -1 ) = 0, f ( v 5 t -1 ) = f ( v 12 t -4 ) = 1, f ( u 9 t -2 ) = 2.
Case 10. n ≡ 9 (mod 12).
Let n = 12 t − 3 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 9 t -2 ) = 2, f ( v 12 t -3 ) = 2.
Case 11. n ≡ 10 (mod 12).
Let n = 12 t − 2 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t ) = 0, f ( v 12 t -2 ) = 2.
Case 12. n ≡ 11 (mod 12).
Let n = 12 t − 1 and t > 0. Define a map f : V ( Gn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 9 t ) = 0, f ( v 12 t -1 ) = 1.
The following table 1 shows that Gn is a Total Mean Cordial graph. □
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The helm Hn is the graph obtained from a wheel by attaching a pendant edge at each vertex of the n -cycle.
Theorem 2.4. Helms Hn are Total Mean Cordial.
Proof . Let V ( Hn ) = { u , ui , vi : 1 ≤ i n } and E ( Hn ) = { ui u i +1 : 1 ≤ i n − 1} ∪ { unu 1 } ∪ { uui , uivi : 1 ≤ i n }. Clearly the order and size of Hn are 2 n + 1 and 3 n respectively.
Case 1. n ≡ 0 (mod 12).
Let n = 12 t and t > 0. Construct a vertex labeling f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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and f ( v 5 t +1 ) = 1.
Case 2. n ≡ 1 (mod 12).
Let n = 12 t + 1 and t > 0. Define a map f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t +1 ) = 0, f ( v 5 t +1 ) = 1, f ( u 12 t ) = f ( u 12 t +1 ) = 1, f ( v 12 t ) = 2 and f ( v 12 t +1 ) = 0.
Case 3. n ≡ 2 (mod 12).
Let n = 12 t + 2 and t > 0. Define a map f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 12 t +1 ) = f ( u 12 t +2 ) = 1 and f ( v 12 t +1 ) = f ( v 12 t +2 ) = 2.
Case 4. n ≡ 3 (mod 12).
The Total Mean Cordial labeling of H 3 is given in figure 3 .
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Let n = 12 t + 3 and t > 0. Define a map f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t +2 ) = 0, f ( v 5 t +2 ) = 1.
Case 5. n ≡ 4 (mod 12).
Let n = 12 t - 8 and t > 0. Define a map f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 12 t -8 ) = 1 and f ( v 12 t -8 ) = 2.
Case 6. n ≡ 5 (mod 12).
The Total Mean Cordial labeling of H 3 is given in figure 4 .
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Let n = 12 t + 5 and t > 0. Define a function f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t +3 ) = 0, f ( v 5 t +3 ) = 1, f ( u 12 t +4 ) = f ( u 12 t +5 ) = 1 and f ( v 12 t +4 ) = f ( v 12 t +5 ) = 2.
Case 7. n ≡ 6 (mod 12).
Let n = 12 t - 6 and t > 0. Define a function f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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Case 8. n ≡ 7 (mod 12).
Let n = 12 t - 5 and t > 0. Define a function f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 12 t -6 ) = f ( u 12 t -5 ) = 1, f ( v 12 t -6 ) = 2 and f ( v 12 t -5 ) = 0.
Case 9. n ≡ 8 (mod 12).
Let n = 12 t - 4 and t > 0. Construct a vertex labeling f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t -1 ) = 0, f ( v 5 t -1 ) = 1, f ( u 12 t -5 ) = f ( u 12 t -4 ) = 1 and f ( v 12 t -5 ) = f ( v 12 t -4 ) = 2.
Case 10. n ≡ 9 (mod 12).
Let n = 12 t - 3 and t > 0. Define f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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Case 11. n ≡ 10 (mod 12).
Let n = 12 t - 2 and t > 0. Define a function f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 5 t ) = 0, f ( v 5 t ) = 1, f ( u 12 t -2 ) = 1 and f ( v 12 t -2 ) = 2.
Case 12. n ≡ 11 (mod 12).
Let n = 12 t - 1 and t > 0. Define a function f : V ( Hn ) → {0, 1, 2} by f ( u ) = 1,
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f ( u 12 t -2 ) = f ( u 12 t -1 ) = 1 and f ( v 12 t -2 ) = f ( v 12 t -1 ) = 2.
The following table 2 shows that Hn is a Total Mean Cordial graph. □
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BIO
Dr. R. Ponraj received his Ph.D in Manonmaniam Sundaranar University, Tirunelveli. He is currently an Assistant Professor at Sri Paramakalyani College, Alwarkurichi, India. His Research interest is in Discrete Mathematics.
Department of Mathematics, Sri Paramakalyani College, Alwarkurichi, Tamil Nadu, India-627412.
e-mail: ponrajmaths@gmail.com
Mr. S. Sathish Narayanan did his M.Phil in St. Johns College, Palayamkottai. He is persuing doctoral research work. His Research interest is in Graph labeling.
Department of Mathematics, Sri Paramakalyani College, Alwarkurichi, Tamil Nadu 627 412, India.
e-mail: sathishrvss@gmail.com
References
Gallian J.A. (2013) A Dynamic survey of graph labeling The Electronic Journal of Combinatorics # Ds6. 16
Harary F. 2001 Graph theory Narosa Publishing house New Delhi
Ponraj R. , Ramasamy A.M.S. , Sathish Narayanan S. Total Mean Cordial labeling of graphs (communicated).
Ponraj R. , Sathish Narayanan S. Total Mean Cordiality ofKcn+2K2(communicated).
Ponraj R. , Ramasamy A.M.S. , Sathish Narayanan S. Total Mean Cordial labeling of some graphs (communicated).