Graceful Labeling for Double Step Grid Graph

Transcription

1 International Jornal of Mathematics And its Applications Volme 3, Isse 1 (015), ISSN: International Jornal of Mathematics Applications And its ISSN: Gracefl Labeling for Doble Step Grid Graph V.J.Kaneria and H.M.Makadia,1 Department of Mathematics, Sarashtra University, Rajkot , India. Department of Mathematics, Government Engineering College, Rajkot , India. Abstract : We investigate a new graph which is called doble step grid graph. We proved that the doble step grid graph is gracefl. We have investigated some doble step grid graph related families of connected gracefl graphs. We proved that path nion of doble step grid graph, cycle of doble step grid graph and star of doble step grid graph are gracefl. Keywords : Gracefl labeling, doble step grid graph, path nion of graphs, cycle of graphs, star of a graph. AMS Sbject Classification : 05C78. 1 Introdction The gracefl labeling was introdced by A. Rosa [1] dring Golomb [] named sch labeling as gracefl labeling, which was called earlier as β valation. In this work we introdce a new graph which is called doble step grid graph and it is denoted by DSt n. We begin with a simple, ndirected finite graph G = (V, E) with V = p vertices and E = q edges. For all terminology and notations we follows Harary [3]. Here are some of the definitions which are sefl in this paper. Definition 1.1. A fnction f is called gracefl labeling of a graph G = (V, E) if f : V {0, 1,..., q} is injective and the indced fnction f : E {1,,..., q} defined as f (e) = f() f(v) is bijective for every edge e = (, v) E. A graph G is called gracefl graph if it admits a gracefl labeling. Definition 1.. Let G be a graph and G 1, G,..., G n, n be n copies of graph G. Then the graph obtained by adding an edge from G i to G i+1 (1 i n 1) is called path nion of G. Definition 1.3 ([4]). For a cycle C n, each vertex of C n is replaced by connected graphs G 1, G,..., G n and is known as cycle of graphs. We shall denote it by C(G 1, G,..., G n ). If we replace each vertex by a graph G, i.e. G 1 = G, G = G,..., G n = G, sch cycle of a graph G is denoted by C(n G). Definition 1.4 ([5]). Let G be a graph on n vertices. The graph obtained by replacing each vertex of the star K 1,n by a copy of G is called a star of G and is denoted by G. 1 Corresponding athor makadia.hardik@yahoo.com (H.M.Makadia)

2 34 Int. J. Math. And its App. Vol.3 No.1 (015)/ V.J.Kaneria and H.M.Makadia Definition 1.5 ([6]). Take P n, P n, P n 1,..., P paths on n, n, n 1, n,..., 3, vertices and arrange them vertically. A graph obtained by joining horizontal vertices of given sccessive paths is known as a step grid graph of size n, where n 3. It is denoted by St n. Obviosly V (St n ) = 1 (n + 3n ) and E(St n ) = n + n. Definition 1.6. Take P n, P n, P n, P n 4,..., P 4, P paths on n, n, n, n 4,..., 4, vertices and arrange them centrally horizontal. where n 0 (mod ), n. A graph obtained by joining vertical vertices of given sccessive paths is known as a doble step grid graph of size n. It is denoted by DSt n. Obviosly V (DSt n ) = n 4 (n + 6) and E(DSt n) = n +3n. In this paper we introdced graceflness of doble step grid graph, path nion of doble step grid graph, cycle of doble step grid graph and star of doble step grid graph. For detail srvey of graph labeling we refer Gallian [7]. Main Reslts Theorem.1. A doble step grid graph DSt n is a gracefl graph, where n 0 (mod ), n. Proof. Let G = DSt n be any doble step grid graph of size n, where n 0 (mod ), n. We mention each vertices of first row like 1,j (1 j n) and nd row like,j (1 j n) and 3 rd row like 3,j (1 j n ) and 4 th row like 4,j (1 j n 4) similarly the last row like n +1,j (1 j ). We see that nmber of vertices in G is V (G) = p = n 4 (n + 6) and the nmber of edges in G is E(G) = q = n +3n. We define labeling fnction f : V (G) {0, 1,..., q} as follows f( 1,1 ) = q; f( i,1 ) = i i, i =, 3,..., n + 1 f( i, ) = q i(i 1), i =, 3,..., n + 1 f( i,j ) = f( i+1,j 1 ) ( 1) i+j, i = 1,, j = i + 1, i +,..., n f( i,j ) = f( i 1,j+ ) + ( 1) j, i = 3, 4,..., n, j = 3, 4,..., n (i ). Above labeling patten give rise a gracefl labeling to the graph G. So G is a gracefl graph. Illstration.. DSt 8 and its gracefl labeling shown in figre.. Figre.: DSt 8, doble step grid graph with n = 8 and its gracefl labeling.

3 Gracefl Labeling for Doble Step Grid Graph 35 Theorem.3. Path nion of finite copies of the doble step grid graph DSt n is a gracefl graph, where n 0 (mod ), n. Proof. Let G be a path nion of r copies for the doble step grid graph DSt n, where n 0 (mod ), n. Let f be the gracefl labeling of DSt n as we mentioned in Theorem.1. In graph G, we see that the vertices V (G) = P = rn 4 (n + 6) and the edges E(G) = Q = rn(n+3) 1. Let k,i,j (i = 1,,..., n + 1, j = 1,,...,min{n, n + 4 i}) be vertices of k th copy of DSt n, k = 1,,..., r. Where the vertices of k th copy of DSt n is p = n 4 (n + 6) and edges of kth copy of DSt n is q = n +3n. Join the vertices k,1,n to k+1,1,1 for k = 1,,..., r 1 by an edge to from the path nion of r copies of doble step grid graph. We define labeling fnction g : V (G) {0, 1,..., Q} as follows g( 1,i,j ) = f( i,j ) if f( i,j ) < q ; = f( i,j ) + (Q q) if f( i,j ) > q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i}; g( k,i,j ) = g( k 1,i,j ) + q 1 if g( k 1,i,j) < Q, = g( k 1,i,j ) q + if g( k 1,i,j) > Q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i}, k =, 3,..., r Above labeling patten give rise a gracefl labeling to given graph G. So path nion of finite copies of the doble step grid graph is gracefl graph. Illstration.4. Path nion of 3 copies of DSt 4 and its gracefl labeling shown in figre.4 Figre-.4: A Path nion of 3 copies of DSt 4 and its gracefl labeling. Theorem.5. Cycle of r copies of doble step grid graph C(r DSt n ) is a gracefl graph, where n 0 (mod ), n and r 0, 3 (mod 4) is gracefl. Proof. Let G = C(r DSt n ) be a cycle of doble step grid graph DSt n. Let f be the gracefl labeling for DSt n as we mentioned in Theorem.1. In graph G, we see that the vertices V (G) = P = rn 4 (n + 6) and the edges E(G) = Q = rn(n+3). Let k,i,j (i = 1,,..., n + 1, j = 1,,...,min{n, n + 4 i}) be the vertices of k th copy of DSt n, k = 1,,..., r. Where the vertices of k th copy of DSt n is p = n 4 (n+6) and edges of k th copy of DSt n is q = n +3n. Join the vertices k,1,n with k+1,1,n for k = 1,,..., r 1 and r,1,n with 1,1,n by an edge to from C(r DSt n ). We define labeling fnction g : V (G) {0, 1,..., Q} as follows g( 1,i,j ) = f( i,j ) if f( i,j ) < q, = f( i,j ) + (Q q) if f( i,j ) > q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i};

4 36 Int. J. Math. And its App. Vol.3 No.1 (015)/ V.J.Kaneria and H.M.Makadia g(,i,j ) = g( 1,i,j ) + (Q q) if g( 1,i,j ) < Q, = g( 1,i,j ) (Q q) if g( 1,i,j ) > Q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i}; g( k,i,j ) = g( k,i,j ) (q + 1) if g( k,i,j ) > Q, = g( k,i,j ) + (q + 1) if g( k,i,j ) < Q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i)}, k = 3, 4,..., r ; g( r +1,i,j ) = g( r 1,i,j ) + (q + ) if g( r 1,i,j ) < Q, = g( r 1,i,j ) (q + 1) if g( r 1,i,j ) > Q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i}; g( r +,i,j ) = g( r,i,j ) + (q + ) if g( r,i,j ) < Q, = g( r,i,j ) (q + 1) if g( r,i,j ) > Q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i}; g( k,i,j ) = g( k,i,j ) (q + 1) if g( k,i,j ) > Q, = g( k,i,j ) + (q + 1) if g( k,i,j ) < Q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i}, k = r + 3, r + 4,..., r. Above labeling patten give rise a gracefl labeling to cycle of r copies for doble step grid graph. Illstration.6. C(4 DSt 6 ) and its gracefl labeling shown in figre.6. Figre.6: A cycle of for copies for DSt 6 and its gracefl labeling. Theorem.7. Star of doble step grid graph (DSt n ) is gracefl,where n 0 (mod ), n. Proof. Let G = (DSt n ) be a star of doble step grid graph DSt n, n 0 (mod ), n. let f be the gracefl labeling for DSt n as we mention in Theorem.1. In graph G, we see that the vertices V (G) = P = p(p + 1) and the edges E(G) = Q = (p + 1)q + p, where p = n 4 (n + 6) and q = n +3n. Let k,i,j (i = 1,,..., n + 1, j = 1,,...,min{n, n + 4 i}) be the vertices of kth copy of DSt n, k = 1,,..., p. Where the vertices of k th copy of DSt n is p = n 4 (n + 6) and edges of kth copy of DSt n is q = n +3n. We mention that central copy of (DSt n ) is (DSt n ) (0) and other copies of (DSt n ) is

5 Gracefl Labeling for Doble Step Grid Graph 37 (DSt n ) (k), k = 1,,..., p. We define labeling fnction g : V (G) {0, 1,..., Q} as follows g( 0,i,j ) = f( i,j ) if f( i,j ) < q, = f( i,j ) + (Q q) if f( i,j ) > q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i}; g( 1,i,j ) = g( 0,i,j ) + p(q + 1) if g( 0,i,j ) < Q, = g( 0,i,j ) p(q + 1) if g( 0,i,j ) > Q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i}; g( k,i,j ) = g( k,i,j ) + (q + 1) if g( k,i,j ) < Q, = g( k,i,j ) (q + 1) if g( k,i,j ) > Q, i = 1,,..., n + 1, j = 1,,..., min{n, n + 4 i}, k =, 3,...,p. We see that difference of vertices for the central copy (DSt n ) (0) of G and its other copies (DSt n ) (k) (1 k p) is precisely following seqence p(q + 1) (q + 1) (p 1)(q + 1). p (q + 1). Using this seqence we can prodce reqired edge label by joining corresponding vertices of (DSt n ) (0) with its other copy (DSt n ) (k) (1 k p) in G. Ths G admits gracefl labeling. Illstration.8. Star graph of DSt 4 and its gracefl labeling shown in figre.8. Figre.8: A star graph of DSt 4 and its gracefl labeling. 3 Conclding Remarks Here we introdced a new graph is called doble step grid graph. Present work contribtes some new reslts. We discssed graceflness of doble step grid graphs, path nion of doble step grid graph, cycle of Doble step graph and star of doble step grid graph. The labeling patten is demonstrated by means of illstrations which provide better nderstanding to derived reslts.

6 38 Int. J. Math. And its App. Vol.3 No.1 (015)/ V.J.Kaneria and H.M.Makadia References [1] A. Rosa, On certain valation of graph, Theory of Graphs (Rome, Jly 1966), Goden and Breach, N. Y. and Paris, (1967), [] S. W. Golomb, How to nmber a graph. In: Graph Theory and Compting (R. C. Read. Ed.) Academic Press. New York, (197), [3] F. Harary, Graph theory, Addition Wesley, Massachsetts, (197). [4] V. J. Kaneria, H. M. Makadia and M. M. Jariya, Gracefl labeling for cycle of graphs, Int. J. of Math. Res., 6()(014), [5] S. K. Vaidya, S. Srivastav, V. J. Kaneria and G. V. Ghodasara, Cordial and 3 eqitable labeling of star of a cycle, Mathematics Today 4(008), [6] V. J. Kaneria and H. M. Makadia, Gracefl labeling for Step Grid Graphs, J. of Adv. in Math., 9(5)(014), [7] J.A.Gallian, A Dynamic Srvey of Graph Labeling,The Electronics Jornal of Combinatorics, 17(014), #DS6.