American Journal of Engineering Research (AJER) 2016 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-5, Issue-10, pp-90-95 www.ajer.org Research Paper Open Access
p - Cordial labeling of Some Cycle Related Graphs 2 D. S. T. Ramesh1, K. Jenita Devanesam2 1
Department of Mathematics, Margoschis College, Nazareth – 628 617, Tamil Nadu, INDIA 2 Department of Mathematics, Pope’s College, Sawyerpuram-628251, Tamil Nadu, India.
p - cordial labeling of a graph G with p vertices is a bijection f: V(G) {1, 2, 3,…, p} 2 1 if f (u) f(v) p 2 and e (0) e (1) 1 . If a graph has p - cordial defined by f (e uv) 2 0 otherwise p - cordial where p represents the nearest integer less than or equal to p labeling then it is called 2 2 2 ABSTRACT: A
f
f
p 2 - co rd ial g raph for n ≥ 3 , excep t fo r n = 4 and the graph p obtained by duplication of an arbitrary vertex by a new edge in cycle C n is 2 - co rd ia l and the graph p obtained by duplication of an arbitrary edge by a new vertex in cycle Cn is 2 -cordial. By a graph we mean In th is p aper we p rove C n is
a finite, undirected graph without multiple edges or loops. For graph theoretic terminology, we refer to Harary [2] and Bondy [1].
Keywords:
p 2
- cordial, cycle.
Definition1.1: Duplication of a vertex v k by a new edge e = vꞌvꞌꞌ in a graph G produces a new graph Gꞌ such that N(vꞌ) = {vk, vꞌꞌ} and N(vꞌꞌ) = {vk,vꞌ}. Definition1.2: Duplication of an edge e = uv by a new vertex w in a graph G produces a new graph Gꞌ such that N(w) = {u, v}. Theorem1.1: Cn is
p 2 - cordial graph for n ≥ 3, except for n = 4.
Proof: Case(i): n = 4m, m = 2, 3, 4, … Let the vertices be u1, u2, …, un.
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n i if i is even and i 2 n i if i is odd and i n 2 Define f(ui) = n 1 i if i is even and n i n 2 n i 1 if i is odd in 2 1 if i n Then the labels of the vertices u1, u2, u3, …, un/2 are respectively n-1, 2, n-3, 4,…n-
n n +1, , 2 2
Hence f (ui ) f (ui 1 ) , i = 1, 2, 3, …,
n -1, are respectively 2
n-3, n-5, n-7, …, 7, 5, 3, 1. Thus the corresponding edge labels are 0, 0, 0, … And the labels of vertices u n , u n 2
1
2
2
n n -1 times, 1, 1, 1, … times. 4 4
, …, un-1, un are respectively
n n n n +2, -1, +4, -3, …, n, 1 2 2 2 2 n n n n And f (ui ) f (ui 1 ) , i = , +1, +2, +3, …, n-3, n-2, n-1, are respectively 2, 3, 5, 7, …, n-5, n-3, n-1. 2 2 2 2 n n Thus the corresponding edge labels are 1,1,1, … times, 0, 0, 0, … times. 4 4 And f (un ) f (u1 ) n 2 The label of the corresponding edge is 0. Hence e f (0) e f (1)
n 2
Case(ii): n = 4m+1, m = 1, 2, 3, … Let the vertices be u1, u2, …, un.
i if i n i if Define f(ui) = n 1 i if i 1 if 1 if Here ef(1) =
n i 2 n and i 2
is even and i is odd i is odd
and
n in 2
n i is even i n 2 in
n 1 n 1 , ef(0) = 2 2
Case (iii): n = 4m + 2, m = 1, 2, 3, …,
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Let the vertices be u1, u2, …, un.
i if n 1 i Define f(ui) = n 1 i if i 1 Hence e f (0) e f (1)
n 1 2 n if i is odd and i 2 n i is even and i n 2 n if i is odd in 2 if i n i
i is even and
n 2
Case (iv): n = 4m+3, m = 1, 2, 3, …, Let the vertices be u1, u2, …, un.
i if ni Define f(ui) = n 1 i i 1 1
e f (0)
n i 1 2 n if i is odd and i 2 n if i is odd and i n 2 n if i is even i n 2 if in i is even and
n 1 n 1 , e f (1) 2 2 u8 = 1 u7 = 8
0
0
0
u1 = 7
0 u2 = 2
u6 = 3
1
1
u5 = 6
u3 = 5
1
1 u4 = 4
The
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p 2 - cordial labeling of c8 Page 92
American Journal of Engineering Research (AJER)
2016
Theorem1.2: The graph obtained by duplication of an arbitrary vertex by a new edge in cycle Cn is
p 2 –
cordial. Proof: Let u1, u2, …, un be vertices and e1, e2, e3, …, en be edges of cycle Cn Without loss of generality we duplicate vn by an edge en+1with end vertices vꞌ and vꞌꞌ. Let the graph so obtained be G. Then |V(G)| = n+2 and |E(G)| = n+3. To define f: V(G) {1, 2, 3, …, n+2} we consider the following cases. Case(i): n = 4m, m = 1, 2, 3, …, Let f(vꞌ) = 2m+1, f(vꞌꞌ) = 2m+2
i if i is even and i 2m 4m 3 i if i is odd and i 2m Define f(ui) = 4m 1 i if i is even and m i n i 2 if i is odd 2m i n 1 if in Here ef(0) = 2m+1, ef(1) = 2m+2 Case(ii): n = 4m+1, m = 1, 2, 3, …, Let f(vꞌ) = 2m+2, f(vꞌꞌ) = 2m+3
i if i is even and i 2m 4m 4 i if i is odd and i 2m Define f(ui) = i 2 if i is even and m i n 4m 2 i if i is odd 2m i n 1 if i n ef(0) = 2m+2, ef(1) = 2m+2 v" = 5
v' = 4
v5 = 1
v1 = 7
v4 = 6
v3 = 3
v2 = 2
Case(iii): n = 4m+2, m = 1, 2, 3, …, Let f(vꞌ) = 2m+1, f(vꞌꞌ) = 2m+4
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i if i is even and i 2m 2 4m 5 i if i is odd and i 2m Define f(ui) = 4m 3 i if i is even and m 2 i n i 2 if i is odd 2m 1 i n 1 if i n ef(0) = 2m+2, ef(1) = 2m+3 Case(iv): n = 4m+3, m = 1, 2, 3, …, Let f(vꞌ) = 2m+2, f(vꞌꞌ) = 2m+3
i if i is even and i 2m 4m 6 i if i is odd and i 2m 1 Define f(ui) = 4m 4 i if i is odd and m 1 i n i 2 if i is even 2m i n 1 if i n ef(0) = 2m+3, ef(1) = 2m+3 Theorem1.3: The graph obtained by duplication of an arbitrary edge by a new vertex in cycle Cn is
p 2 -
cordial expect for n = 3. Proof: Let u1, u2, …, unbe vertices and e1, e2, e3, …, en be edges of cycle Cn Without loss of generality we duplicate the edge unu1 by a vertex vꞌ.
V(G) n 1 and E (G) n 2 . To define f: V(G) {1, 2, 3, …, n+1}we consider the following cases. Then
Case(i): n = 4m, m = 1, 2, 3, …, p = 4m+1, q = 4m+2 Let f(vꞌ) = 2m+1
i if i is even and i 2m 4m 2 i if i is odd and i 2m Define f (ui ) 4m 1 i if i is even and m i n i 1 if i is odd 2m i n 1 if i n ef(0) = 2m+1, ef(1) = 2m+1 Case(ii): n = 4m+1, m = 1, 2, 3, …, p = 4m+2; q = 4m+3
i if i is even and i 2m 4m 3 i if i is odd and i 2m Define f (ui ) 4m 2 i if i is odd and m i n i 1 if i is even 2m i n 1 if i n ef(0) = 2m+1, ef(1) = 2m+2 Case(iii): n = 4m+2, m = 1, 2, 3, …, p = 4m+3; q = 4m+4. Let f(vꞌ) = 2m+2
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i if i is even and i 2m 4m 3 i if i is odd and i 2m Define f (ui ) 4m 3 i if i is even and m i n i 2 if i is odd 2m i n 1 if i n ef(0) = 2m+2, ef(1) = 2m+2 v' = 4 u6 = 1
u1 = 6
u5 = 7
u2 = 2
u4 = 3
u3 = 5
Case(iv): n = 4m+3, m = 1, 2, 3, …, p = 4m+4; q = 4m+5 Let f(vꞌ) = 2m+2
i if i is even and i 2m 4m 4 i if i is odd Define f (ui ) i 2 if i is even and 2m i n 1 if i n ef(0) = 2m+2, ef(1) = 2m+3.
REFERENCES [1]. [2]. [3].
J. A. Bondy and U. S. R. Murty, Graph Theory with applications, Macmillan press, London (1976) J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of combinatorics (2014). F. Harary, Graph Theory Addison – Wesley, Reading Mars., (1968)
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