International Journal of Mathematics Research. ISSN 0976-5840 Volume 5, Number 1 (2013), pp. 19-26 © International Research Publication House http://www.irphouse.com Strong Edge Graceful Labeling of Windmill Graphs Dr. M. Subbiah VKS College Of Engineering& Technology Desiyamangalam, Karur - 639120 [email protected]. Abstract A (p, q) graph G is said to have strong edge graceful labeling if there 3q exists an injection f from the edge set to 1,2, ... so that the 2 induced mapping f+ defined on the vertex set given by f x fxy xy EG mod 2p are distinct. A graph G is said to be strong edge graceful if it admits a strong edge graceful labeling. In this paper we investigate strong edge graceful labeling of Windmill graph. (n) Definition: The windmill graphs Km (n >3) to be the family of graphs consisting of n copies of Km with a vertex in common. (n) Theorem: 1. The windmill graph K4 is strong edge graceful for all n 3 when n is even. (n) Proof: Let {v1, v2, v3, ..., v3n, } be the vertices of K4 and {e1, e2, e3, ...,e3n- (n) 1, e3n, , f1, ,f2, ,f3, . .f3n-1, f3n. } be the edges of K4 which are denoted as in the following Fig. 1. 20 Dr. M. Subbiah . v e 3 3 n n -1 . -1 . v . 3 n f . 3 n -2 f 3 e n . 3 -1 n e 3 2 n n -2 f v v 3 3n 0 2 v 3 . n-2 f v 1 . 2 f 2 f 3 f . f 5 4 n 1 n e v 3 v 3 e 1 2 2 2 e . v 4 v 1 4 v 5 e v 3 3 e e 6 5 (n) Fig. 1: K4 with ordinary labeling (n) We first label the edges of K4 as follows: f (fi ) = i 1 i f (fi ) = 3n+1+ i +1 i 3n f (ei ) = 3n+1- i 1 i f (ei ) = 6n+2 - i +1 i 3n Then the induced vertex labels are: f v0 = 0 + f ( vi ) = 6n+2 - i 1 i + f ( vi ) = 3n+1- i +1 i 3n Strong Edge Graceful Labeling of Windmill Graphs 21 (n) Clearly, the vertex labels are all distinct. Hence The windmill graph K4 is strong edge graceful for all n 3 when n is even. (4) (8) The SEGL of K4 , K4 are illustrated in Fig.2, Fig.3, respectively. 4 17 18 19 6 5 2 22 22 21 20 15 23 6 7 24 0 5 1 20 9 16 4 14 25 8 1 2 3 3 21 25 24 12 11 10 23 (4) Fig.2. K4 with SEGL 22 Dr. M. Subbiah 7 32 33 4 10 30 35 9 8 12 29 5 34 36 37 31 42 44 43 41 40 11 6 45 39 2 46 38 40 27 47 12 13 0 15 1 48 11 38 28 49 10 26 1 14 9 3 2 39 48 8 43 3 7 4 5 6 18 24 42 23 49 21 16 45 46 22 17 47 41 20 19 44 (8) Fig.3. K4 with SEGL (n) Theorem: 2. The windmill graph K4 is strong edge graceful for all n 3 when n 1(mod 4). (n) Proof: Let {v1, v2, v3, ..., v3n, } be the vertices of K4 and {e1, e2, e3, ...,e3n- (n) 1, e3n, f1, ,f2, ,f3, . .f3n-1, f3n. } be the edges of K4 which are denoted as in the above Fig. 1. (n) We first label the edges of K4 as follows: f ( fi ) = i 1 i 3n-1 f ( f3n ) = 6n f (ei ) = 6n- i 1 i 3n Strong Edge Graceful Labeling of Windmill Graphs 23 Then the induced vertex labels are: + f ( v0 ) = + f ( vi ) = 6n – 2 - i 1 i 3n-3 + f ( v3n-2 ) = 6n + f ( vi ) = 6n – 2 - i 3n - 1 i 3n (n) Clearly, the vertex labels are all distinct. Hence The windmill graph K4 is strong edge graceful for all n ≥3 when n 1(mod 4). (5) (9) The SEGL of K4 , K4 are illustrated in Fig.4, Fig.5. respectively. 16 18 19 17 18 20 14 21 16 21 13 12 11 10 19 13 9 14 8 7 23 15 7 22 17 30 6 1 20 30 26 5 2 24 29 3 4 26 24 28 27 23 25 27 22 25 (5) Fig.4. K4 with SEGL 24 Dr. M. Subbiah 34 31 36 37 37 34 39 36 35 33 38 39 40 32 41 35 38 31 33 42 28 29 18 17 16 42 40 20 19 15 14 22 21 13 44 30 23 12 43 32 24 11 30 26 13 10 41 25 9 28 26 8 45 29 54 7 47 1 25 2 6 27 54 3 4 5 45 50 44 53 50 48 46 47 51 43 52 51 48 49 49 46 (9) Fig.5. K4 with SEGL (n) Theorem: 3. The windmill graph K4 is strong edge graceful for all n 3 when n 3(mod 4). (n) Proof: Let {v1, v2, v3, ..., v3n, } be the vertices of K4 and {e1, e2, e3, ...,e3n- (n) 1, e3n, , f1, ,f2, ,f3, . .f3n-1, f3n. } be the edges of K4 which are denoted as in the above Fig. 1. (n) We first label the edges of K4 as follows: f (fi ) = i 1 i 3n f (ei ) = 6n+1 i 1 i 3n Strong Edge Graceful Labeling of Windmill Graphs 25 Then the induced vertex labels are: + f ( v0) = + f ( vi ) = 6n - i 1 i 3n (n) Clearly, the vertex labels are all distinct. Hence The windmill graph K4 is strong edge graceful for all n 3 when n 3(mod 4). (3) (7) The SEGL of K4 , K4 are illustrated in Fig.6, Fig.7. respectively. 9 10 11 12 11 10 16 14 9 8 7 1 17 6 15 13 2 15 5 12 18 5 16 3 4 14 17 13 (3) Fig.6. K4 with SEGL 26 Dr. M. Subbiah 27 24 28 29 26 29 25 28 30 30 31 25 32 33 27 15 14 32 26 16 13 22 17 12 18 11 23 31 19 11 10 35 21 9 36 24 20 34 22 21 8 1 6 7 23 2 3 4 5 33 34 35 40 39 38 42 37 41 37 41 38 40 39 36 (7) Fig.7. 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