SQUARE DIFFERENCE 3-EQUITABLE LABELING FOR SOME GRAPHS. Sattur , TN, India. Sattur , TN, India.

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1 International Journal of Science, Engineering Technology Research (IJSETR), Volume 5, Issue, March 2016 SQUARE DIFFERENCE -EQUITABLE LABELING FOR SOME GRAPHS R Loheswari 1 S Saravana kumar 2 1 Research Scholar, Department of Mathematics, Sri SRNM College, Sattur , TN, India 2 Department of Mathematics, Sri SRNM College, Sattur , TN, India Abstract: A square difference -equitable labeling of a graph G with vertex set V is a bijection f from V to {1, 2,, V } such that if each edge uv is assigned the label 1 if [f(u)] 2 [f(v) 2 ] 1(mod 4), the label 0 if [f(u)] 2 [f(v)] 2 0(mod 4) the label 1 if [f(u)] 2 [f(v)] 2 1(mod 4), then the number of edges labeled with i the number of edges labeled with j differ by atmost 1 for 1 i, j 1 If a graph has a square difference -equitable labeling, then it is called square difference -equitable graph In this paper, we investigate the square difference -equitable labeling behaviour of comb, triangular snake, crown, star, wheel Keywords: Square difference -equitable labeling, square difference -equitable graphs AMS Subject Classification (2010): 05C78 1 INTRODUCTION All graphs considered here are finite, simple undirected Gallian has given a dynamic survey of labeling [1] For graph theoretic terminologies notations we follow Harary [2] The concept of - equitable labeling was introduced by SK Vaidya NH Shah in the year 2012[7] A ternary vertex labeling of a graph G is called a -equitable labeling if v f (i) v f (j) 1 e f (i) e f (j) 1 for all 1 i, j 1 A graph G is -equitable if it admits - equitable labeling The concept of square difference labeling was introduced by J Shiama in the year of [, 4, 5, 6] The concept of square difference - equitable labeling of pahs cycle was introduced by S Murugesan J Shiama in the year of 2015[8] 2 PRELIMINARIES Definition 21 Let G = (V, E) be a graph A mapping f : V (G) { 1, 0, 1} is called ternary vertex v of G f(v) is called the label of the vertex v of G under f For an edge e = uv, he induced edge labeling is given by f* : E(G) { 1, 0, 1} Let v f ( 1), v f (0), v f (1) be the number of vertices of G having labels 1, 0, 1 respectively under f e f ( 1), e f (0), e f (1) be the number of edges having labels 1, 0, 1 respectively under f * Definition 22 A ternary vertex labeling of a graph G is called a -equitable labeling if v f (i) v f (j) 1 e f (i) e f (j) 1 for all 1 i, j 1 A graph G is -equitable if it admits -equitable labeling Definition 2 A square difference - equitable labeling of a graph G wih vertex set V (G) is a bijection f : V (G) {1, 2,,, V } such that the induced edge labeling f*: E(G) { 1, 0, 1} is defined by 680

2 International Journal of Science, Engineering Technology Research (IJSETR), Volume 5, Issue, March 2016 f*(e=uv) 1 [f(u)]2 [f(v)]2 1(mod 4) = 0 [f(u)]2 [f(v)]2 0(mod 4) 1 [f(u)]2 [f(v)]2 1(mod 4) Definition 24 The corona G 1 ʘG 2 of two graphs G 1 G 2 is defined as the graph G obtained by taking one copy of G 1 (which has p1 points) p1 copies of G 2 then joining the i th copy of G 1 to every point in the i th copy of G 2 The graph P n ʘ K 1 is called a comb Definition 25 A Triangular Snake is obtained from a path v 1, v 2, v n joining v i v i+1 to a new vertex w i for 1 i n 1 That is, every edge of a path is replaced by a triangle C Definition 26 C n + is a graph obtained from G by attaching a pendent vertex from each vertex of the graph Cn is called Crown Definition 27 The graph K 1,r, r 1 is called a star at the vertex has degree r is called center Definition 28 A graph C n +K 1 is called a wheel with n spokes is denoted by W n MAIN RESULTS Theorem 1 The comb P n + admits square difference -equitable labeling Proof Let G be a comb obtained from a path P n = v 1, v 2,, v n by joining a vertex u i to v i The labeling of P 1 + P 2 + are given as follows If n, we consider the following cases Case-1: n 1(mod 6) f(v 2 ) = f(v ) = 1 for 1 i n 7 6, f(v 6i+ ) = 6i + f(v n 2 ) = n f(v n 1 ) = n 2 f(v n ) = n 1 [f(v 1 )] 2 [f(v 2 ] 2 1(mod4) ) f(v 1 v 2 ) =1 [f(v 2 )] 2 [f(v ] 2 0(mod4) ) f(v2v) = 0 [f(v )] 2 [f(v 4 ] 2 1(mod4) f(v v 4 )= 1, for 1 i n 7 6, [f(v 6i 2 ] 2 [f(v 6i 1 )] 2 1(mod4) f*(v 6i v 6i 1 ) = 1 [f(v 6i 1 ] 2 [f(v 6i )] 2 0(mod4) f*(v 6i-1 v 6i ) = 0 [f(v 6i ] 2 [f(v 6i+1 )] 2 1(mod4) f*(v 6i v 6i+1 ) = 1 [f(v 6i+1 ] 2 [f(v 6i+2 )] 2 0(mod4) f*(v 6i+1 v 6i+2 ) = 0 [f(v 6i+2 )] 2 [f(v 6i+ )] 2 1(mod4) f*(v 6i+2 v 6i+ ) = 1 [f(v 6i+ )] 2 [f(v 6i+4 )] 2 1(mod4) f*(v 6i+ v 6i+4 ) = 1 [f(v n )] 2 [f(v n 2 ] 2 1(mod4) f*(v n v n 2 ) = 1 [f(v n 2 )] 2 [f(v n 1 ] 2 0(mod4) f*(v n 2 v n 1 )=0 [f(v n 1 )] 2 [f(v n) ] 2 1(mod4) f*(v n 1 v n ) = 1 f(u j ) = n + j, 1 j 5 681

3 International Journal of Science, Engineering Technology Research (IJSETR), Volume 5, Issue, March 2016 f(u j ) = n + j + 1 j 0(mod ) n + j 1 j 1(mod ) j n n + j j 2(mod ) 2n +1 2n 1 e f ( 1) = e f (0) = e f (1) = Case-2: n 2(mod6) f(v 2 ) = f(v ) = 1 for 1 i n 8 f(v 6i+ ) = 6i + f(v n ) = n 1 f(v n 2 ) = n f(v n 1 ) = n 2 f(v n ) = n n + j j 0, 1, 2, 5(mod 6) f(u j ) = n + j + 1 j (mod 6) n + j 1 j 4(mod 6) e f ( 1) = e f (0) = e f (1) = 2n 1 Case-: n (mod6) f(v 2 ) = f(v ) = 1 for 1 i n 9 f(v 6i+ ) = 6i + f(v n 4 ) = n 2 f(v n ) = n 4 f(v n 2 ) = n f(v n 1 ) = n 1 f(v n ) = n n + j + 1 f(u j ) = for j = 1, 2 n + j 1 f(u j ) = n + j, j n e f ( 1) = e f (1) = 2n e f (0) = 2n 1 4 Case-4: n 4(mod6) f(v 2 ) = f(v ) = 1 for 1 i n 4 f(v 6i+ ) = 6i + f(u j ) = n + j j 0, 1, 4, 5(mod 6) n + j + 1 j 2(mod 6) n + j 1 j (mod 6) 2n 1 e f ( 1) = e f (1) = Case-5: n 5(mod6) f(v 2 ) = f(v ) = 1 for 1 i n 5 f(v 6i+ ) = 6i + f(v n ) = n e f (0) = 2n

4 International Journal of Science, Engineering Technology Research (IJSETR), Volume 5, Issue, March 2016 f(u j ) = n + j, 1 j n e f ( 1) = e f (0) = e f (1) = 2n Case-6: n 0(mod6) f(v 2 ) = f(v ) = 1 f(v i ) = i, 5 i n n + j j 1,2,,4(mod 6) f(u j ) = n + j + 1 j 5(mod 6) n + j 1 j 0(mod 6) 1 2n +1, Thus in all cases, e f (i) e f (j) 1 for all 1 i, j 1 therefore P + n is a square difference -equitable labeling graph Illustration 1 The Square difference - equitable labeling of P 12 + is shown below Theorem 2 The Triangular Snake Tn admits a square difference -equitable labeling Proof Let T n be a path v 1, v 2,, v n joining v i v i+1 to a new vertex w i f(v i ) = i, 1 i n f(w i ) = n + i, 1 i n 1 e f ( 1) = e f (1) = n 1 e f (0) = n 2 Thus e f (i) e f (j) 1 for all 1 i, j 1 therefore T n is a square difference - equitable labeling Illustration 2 The Square difference - equitable labeling of T 7 is shown below Theorem The Crown C + n admits a square difference -equitable labeling except n (mod6) Proof Let v 1 v 2 v v n v 1 be the cycle C n Let u i be the vertex which is adjacent to v i, 1 i n Case-1: n 1(mod6) f(v 2 ) = f(v ) = 1 f(v 4 ) = 4 for1 i n 7 f(v 6i+ ) = 6i + f(v n 2 ) = n f(v n 1 ) = n 2 f(v n ) = n 1 f(v 1 u 1 ) = 2n f(v 2 u 2 ) = n + i 1 n + i f(v i u i ) = n + i 2 i n 1 f(v n u n ) = n + i 1 Case-2: n 2(mod6) f(v 2 ) = 1 f(v ) = 4 f(v 4 ) = 5, 2n 68

5 International Journal of Science, Engineering Technology Research (IJSETR), Volume 5, Issue, March 2016 f(v 5 ) = for1 i n 8 f(v 6i ) = 6i + 2 f(v 6i+2 ) = 6i + 1 f(v 6i+ ) = 6i + f(v 6i+5 ) = 6i + 5 f(v n 2 ) = n f(v n 1 ) = n 2 f(v n ) = n 1 f(v 1 u 1 ) = 2n n + i 1 i 4, 5(mod 6) f(v i u i ) = n + i i 0, 2(mod 6) n + i 2 i 1, (mod 6) f(v n u n ) = n + i 1, 2n Case-: n 4(mod6) f(v 2 ) = f(v ) = 1 f(v 4 ) = 4 for 1 i n 4 f(v 6i+ ) = 6i + f(v i u i ) = n + i i 0, 1, 2, 5(mod 6) n + i + 1 i (mod 6) n + i 1 i 4(mod 6) Case-4: n 5(mod6) f(v 2 ) = 1 f(v ) = 4, 2n f(v 4 ) = 5 f(v 5 ) = for1 i n 5 f(v 6i ) = 6i + 2 f(v 6i+2 ) = 6i + 1 f(v 6i+ ) = 6i + f(v 6i+5 ) = 6i + 5 f(v 1 u 1 ) = 2n f(v n u n ) = n + 2 n + i 1 i 2, 4(mod 6) f(v i u i ) = n + i i 0, 5(mod 6) n + i + 1 i 1, (mod 6), 2n Case-5: n 0(mod6) f(v 2 ) = f(v ) = 5 f(v 4 ) = 4 f(v 5 ) = 1 f(v 6 ) = 6 for 1 i n f(v 6i+2 ) = 6i + 1 f(v 6i+ ) = 6i + 2 f(v 6i+5 ) = 6i + 5 f(v 6i+6 ) = 6i + 6 n + i i 1, 2,, 4(mod 6) f(v i u i ) = n + i + 1 i 5(mod 6) n + i 1 i 0(mod 6), 2n Thus in above cases, e f (i) e f (j) 1 for all 1 i, j 1 therefore C n is a square difference -equitable labeling graph Case-6: n (mod6) We see that e f (i) e f (j) =0 or 1 but e f (i) e f (j) =2 684

6 International Journal of Science, Engineering Technology Research (IJSETR), Volume 5, Issue, March 2016 Thus C n + is not square difference -equitable labeling Hence C n + is square difference -equitable labeling except n (mod6) Illustration The square difference - equitable labeling of C 8 + is shown below 1 i, j 1 Case-() : n = 9 We see that e f ( 1) e f (0) =2 hence K 1,9 is not square difference -equitable labeling Case-(4) : n 11 We see that e f ( 1) e f (0) Thus K 1,n is not square difference - equitable labeling for n 9 except n = 10 Illustration 4 The square difference - equitable labeling of K 1,8 is shown below Theorem 4 The Star graph K 1,n is not square difference -equitable labeling for n 9 except n = 10 Proof Let v be the central vertex let v 1, v 2,, v n be the end vertices of the star K 1,n Case-(1) : n = 1, 2, The square difference -equitable labeling of K 1,1,K 1,2 K 1, are given as follows Case-(2) : 4 n 8 n = 10 Assign the label 4 to the vertex v the remaining labels to the vertices v 1, v 2,, v n We see that e f (i) e f (j) 1 for all Theorem 5 The wheel graph W n admits a square difference -equitable labeling if n 1, 2(mod6) Proof Let W n = C n + K 1 Let C n be a cycle u 1, u 2,, u n, u 1 V (K 1 ) = {u} If n 5, we consider the following cases Case-(1) : n 1(mod6) f(u 1 ) = 1 f(u 2 ) = f(u ) = 2 f(u 4 ) = 5 for1 i n 7 6, Subcase-(i): i is odd f(u 6i 1 ) = 6i 2 f(u 6i ) = 6i + 1 f(u 6i+1 ) = 6i f(u 6i+2 ) = 6i + 2 f(u 6i+ ) = 6i

7 International Journal of Science, Engineering Technology Research (IJSETR), Volume 5, Issue, March 2016 f(u 6i+4 ) = 6i + Subcase-(ii): i is even f(u 6i 1 ) = 6i 2 f(u 6i ) = 6i + 1 f(u 6i+1 ) = 6i f(u 6i+2 ) = 6i + 4 f(u 6i+ ) = 6i + 2 f(u 6i+4 ) = 6i + f(u n 2 ) = n f(u n 1 ) = n f(u n ) = n 1 f(u) = n + 1 Case-(2) : n 2(mod6) Subcase-(i) : n = 8 f(u 1 ) = 2 f(u 2 ) = f(u ) = 1 f(u 4 ) = 4 f(u 5 ) = 7 f(u n 2 ) = n f(u n 1 ) = n 2 f(u n ) = n + 1 Subcase-(ii) : n > 8 f(u 1 ) = 2 f(u 2 ) = 1 f(u ) = f(u 4 ) = 4 f(u 5 ) = 7 for 1 i n 8 6, Subcase-(a) : i is odd f(u 6i ) = 6i 1 f(u 6i+1 ) = 6i f(u 6i+2 ) = 6i + f(u 6i+ ) = 6i + 4 f(u 6i+4 ) = 6i + 8 f(u 6i+5 ) = 6i + 7 Subcase-(b):i is even f(u 6i ) = 6i 1 f(u 6i+1 ) = 6i, 2n f(u 6i+2 ) = 6i + f(u 6i+ ) = 6i + 4 f(u 6i+4 ) = 6i + 7 f(u 6i+5 ) = 6i + 8 f(u n 2 ) = n f(u n 1 ) = n 2 f(u n ) = n + 1 f(u) = 8, 2n Case-() : n 0,, 4, 5(mod6) We see that e f (i) e f (j) =0 or 1 but e f (i) e f (j) 2 Thus W n is not square difference -equitable labeling Hence W n is square difference -equitable labeling if n 1, 2(mod6) Illustration 5 The square difference - equitable labeling of W 8 is shown below References [1] JA Gallian, A dynamic survey of graph labeling, Electronic Journal of Combinatorics, 17(2010), DS6 [2] F Harary, Graph Theory, Addition- Wesley, Reading, Mass,

8 International Journal of Science, Engineering Technology Research (IJSETR), Volume 5, Issue, March 2016 [] J Shiama, Square sum labeling for some middle total graphs, International Journal of Computer Applications ( ) volume 7-No4 January 2012 [4] J Shiama, Square difference labeling for some graphs, International Journal of Computer Applications ( ) volume 44-No 4 April 2012 [5] J Shiama, Some Special types of Square difference graphs, International Journal of Mathematical archives -(6), 2012, ISSN [6] J Shiama, Square difference labeling for some path, fan, gear graphs, International Journal of Scientific Engineering Research volume 4, issues, March -201, ISSN [7] SK Vaidya, NH Shah, -Equitable Labeling for Some Star Bistar Related Graphs, International Journal of Mathematics Scientific Computing, (ISSN: ), vol 2, No 1, 2012 [8] S Murugesan, J Shiama, Square Difference -Equitable Labeling of Paths Cycles, International Journal of Computer Applications ( ), Volume 12- No 17, August