Super edge-magic total labeling of a tree

Transcription

1 Super edge-magi total labeling of a tree K. Ali, M. Hussain, A. Razzaq Department of Mathematis, COMSATS Institute of Information Tehnology, Lahore, Pakistan. {akashifali, mhmaths, asimrzq}@gmail.om Abstrat. An edge-magi total labeling of a graph G is a one-toone map λ from V (G) E(G) onto the integers {,,, V (G) E(G) } with the property that, there is an integer onstant suh that λ(x) + λ(xy) + λ(y) = for any xy E(G). If λ(v (G)) = {,,, V (G } then edge-magi total labeling is alled super edgemagi total labeling. In this paper we investigate super edge-magi total labeling on w-trees. Keywords : Super edge-magi total labeling, w-tree. Introdution All graphs in this paper are finite, simple, planar and undireted. Graph G has the vertex-set V (G) and edge-set E(G). A general referene for graphtheoreti ideas an be seen in [5]. A labeling (or valuation) of a graph is a map that arries graph elements to numbers (usually to positive or non-negative integers). In this paper the domain will usually be the set of all verties and edges and suh labelings are alled total labelings. Some labelings use the vertex-set only, or the edge-set only, and we shall all them vertex-labelings and edge-labelings respetively. Other domains are possible. The most omplete reent survey of graph labelings an be seen in [0]. There are many types of graph labelings, for example harmonius, ordial, graeful and antimagi. In this paper, we fous on one type of labeling alled edge-magi total labeling. An edge-magi total labeling of a graph G is a one-to-one map λ from V (G) E(G) onto the integers {,,, V (G) E(G) } with the property that, there is an integer onstant suh that λ(x)+λ(xy)+λ(y) = for any xy E(G). An edge-magi total labeling λ of a graph G is alled a super edge-magi total labeling if λ(v (G)) = {,,, V (G }. The subjet of edge-magi total labelings of graphs has its origin in the work of Kotzig and Rosa [, 3], on what they alled magi valuations

2 K. Ali, M. Hussain, A. Razzaq of graphs. A more restritive form of edge-magi labeling was defined by Enomoto et al. in [7] as super edge-magi total labelings whih is an equivalent onept of strongly indexable labelings defined before by Arharya and Hedge [3]. A number of lassifiation studies on edge-magi total graphs has been intensively investigated. A part of these studies results inludes Every yle C n is super edge-magi total if and only if n is odd [7]. K m,n is super edge-magi total if and only if m = or n = [7]. K n is super edge-magi total if and only if n =,, or 3 [7]. nk is super edge-magi total if and only if n is odd [6]. P n is super edge-magi total if and only if n is not or 3 [9]. The friendship graph onsisting of n triangles is super edge-magi total if and only if n is 3, 4, 5 or 7 [4]. np 3 is super edge-magi total for even n 4 [5], and np 3 is super edge-magi total for n odd [9]. The fan F n is super edge-magi total if and only if n 6 [8]. If G is a (super) edge-magi bipartite or tripartite graph, and n is odd, then ng is (super) edge-magi [9]. Probably the most famous onjeture in the world of super edge-magi total labeling laims that every tree is super edge-magi total [6]. To our knowledge this onjeture remains open. Definition. Let G be a graph with vertex and edge sets as follows: V (G) = {(,, b, w, d) (x, x,..., x n ) (y, y,..., y n )} E(G) = { x, x,..., x n } { y, y,..., y n } { b, w} { w, d}, we shall all it w-graph and it shall be denote by W n. Definition. Suppose that we have k opies of w-graphs W n, W n,..., W nk, n n... n k. A w-tree W T (n, n, n 3,..., n k ; k) is a tree obtained by taking a new vertex a and joining it with {d ni : i k}, where d ni is the d vertex of w-graph W ni. In this paper we present super edge-magi total labelings for w-trees as well as for disjoint union of isomorphi and non isomorphi opies of w-trees. Main Results Before introduing our main theorems, let us onsider the following lemma found in [8] that gives a neessary and suffiient ondition for a graph to be super edge-magi total.

3 Super edge-magi total labeling of a tree 3 Lemma A graph G with v verties and e edges is super edge-magi total if and only if there exists a bijetive funtion f : V (G) {,,, v} suh that the set S = {f(x) + f(y) xy E(G)} onsists of e onseutive integers. In suh a ase, f extends to a super edge-magi total labeling of G with magi onstant = v + e + s, where s = min(s). In the following setions we present super edge-magi total labelings of w-trees and forests onsisting of disjoint unions of w-trees.. Super edge-magi total labelings of w-trees Theorem Every w-tree G = W T (n, n, n 3,..., n k ; k) admits a super edge-magi total labeling if n i k when k is even and n i k when k is odd, n n... n k. Proof. If k = then the resulting tree is a aterpillar, and every aterpillar is super edge-magi []. Therefore we assume that k 3. If v = V (G), e = E(G) then v = 5k + + k p= n p and e = v. We denote the vertex and edge sets of G as follows: V (G) = {a} {b i : i k} {w i : i k} {d i : i k} { ij : i k, j =, } {x l i, yl i : i k, l n i} E(G) = {b i i : i k} {w i i : i k} {w i i : i k} {d i i : i k} {ad i : i k} { i x l i, iy l i : i k, l n i.} Before formulating the labeling, let s = k and α i = i n p i N. p= Now, we define labeling λ : V {,,..., v} as follows: k for even k, s = for odd k λ(a) = v k. λ(a) + i (s + ), for i s, λ( i ) = λ( ) + i (s ), for + s i k,

4 4 K. Ali, M. Hussain, A. Razzaq λ( i ) +, for i s, λ( i ) = λ( i ), for + s i k, α i + 3i, for i s, λ(b i ) = α i + 3i, for + s i k, λ(b i ) + n i +, for i s, λ(w i ) = λ(b i ) n i, for + s i k, λ(w i ) + n i + i s +, for i s, λ(d i ) = λ(w i ) n i + i s 3, for + s i k, For i s, For + s i k λ(x l i) = λ(b i ) + l, for l n i λ(y l i) = {λ(w i ) + l}\{λ(d i )}, for l n i +. λ(x l i) = {λ(b i ) l}, for l n i λ(y l i) = {λ(w i ) l}\{λ(d i )}, for l n i The set of all edge-sums generated by the above formula forms a onseutive integer sequene v t s +, v t s +, v t s + 3,..., v + e t s, where t = k. Therefore, by Lemma λ an be extended to a super edge-magi total labeling and we obtain the magi onstant = v + e + s = v + e + v t s + = 3v t s. Using theorem. found in [9], we obtain the following immediate orollary of theorem : Corollary Consider a w-tree G = W T (n, n, n 3,..., n k ; k) suh that n i k when k is even and n i k when k is odd, where k 3 and n n... n k. If m is an odd integer, then mg is super edge-magi.

5 Super edge-magi total labeling of a tree 5. Super edge-magi total labelings on m isomorphi opies of w-trees Theorem For k 3 and m, G = mw T (n, n, n 3,..., n k ; k) admits a super edge-magi total labeling if n i mk, i k, and n n n 3... n k. Proof. If v = V (G) and e = E(G) then v = 5mk+m+m k n p and e = v m. p= We denote the vertex and edge sets of G as follows: V (G) = {a r : r m} {b r i : i k, r m} {wi r : i k, r m} {d r i : i k, r m} { r ij : i k, r m, j =, } {x lr i, ylr i : i k, m h, l n i } E(G) = {a r d r i, r i dr i, r i wr i, r i wr i, r i br i : i k, r m} { r i xlr i, r i ylr i : i k, r m,, l n i } Now, we define labeling λ : V {,,..., v} as follows: λ(a r ) = v + r m, r m Throughout the following labeling, we onsider: i k, r m, j =, and α i = i n p i N. p= λ( r ij) = λ(a ) mk + (j ) + (i ) + k(r ), λ(b r i ) = 3i + (r )(α k + 3k) + α i, λ(w r i ) = λ(b r i ) + n i +, λ(d r i ) = λ(w r i ) + λ( r i) + n i + λ(a r ), λ(x lr i ) = λ(b i ) + l, for l n i λ(y lr i ) = {λ(w i ) + l}\{λ(d i )}, for l n i +. The set of all edge-sums generated by the above formula forms the onseutive integer sequene m(3k +α k )+, m(3k +α k )+3,, m(3k +α k )+e+. Therefore, by Lemma λ an be extended to a super edge-magi total labeling and we obtain the magi onstant = v +e+s = 3v mk m+. For example Figure shows a super edge-magi total labeling of W T (, 0, 0; 3).

6 6 K. Ali, M. Hussain, A. Razzaq Fig.. Super edge-magi total labeling of W T (, 0, 0; 3) with = 455.

7 Super edge-magi total labeling of a tree 7.3 Super edge-magi total labelings of m non-isomorphi opies of w-trees Theorem 3 For m, k q k q+ 3, and q m, G = m q= W T (nq, nq, nq 3,..., nq k q ; k q ) admits a super edge-magi total labeling, where n q i mk q for i k q. Proof. If v = V (G) and e = E(G) then, v = 5 m q= k q + m q= ( k q p= nq p) + m and e = v m. We denote the vertex and edge sets of G as follows: V (G) = {a r : r m} {b r i : i k r, r m} {wi r : i k r, r m} {d r i : i k r, r m} { r ij : i k r, r m, j =, } {x rl i, yrl i : i k r, r m, l n i } E(G) = {b r i r i : i k r, r m} {wi rr i : i k r, r m} {wi rr i : i k r, r m} {d r i r i : i k r, r m} {a r d r i : i k r, r m} { r i xrl i, r i yrl i : i k r, r m, l n i } Now, we define labeling λ : V {,,..., v} as follows: λ(a r ) = v + r m, For i k r, r m, j λ( r ij) = λ(a ) m p= r λ(b r i ) = 3i + βi r + ( where β r i = m q= ( i p= nq p). r m r k p + (j ) + (i ) + k p, k q q= p= λ(w r i ) = λ(b r i ) + n r i +, p= r n q p) + 3 k q, i k r, q= i k r, r m λ(d r i ) = λ(wr i ) + λ(r i ) + nr i + λ(ar ), i k r, r m λ(x rl i ) = λ(b r i ) + l, for l n i, i k r, r m

8 8 K. Ali, M. Hussain, A. Razzaq λ(y rl i ) = {λ(w i ) + l}\{λ(d i )}, for l n i, i k r, r m. The set of all edge-sums generated by the above formula forms the onseutive integer sequene v m p= k p m +, v m p= k p m + 3, v m p= k p m+4,, v m p= k p m+. Therefore, by Lemma, λ an be extended to a super edge-magi total labeling and we obtain the magi onstant = v + e + s = 3v m p= k p m +. In Figure and 3 we show W T (n, n, n 3; 3) W T (n, n, n 3; 3), and a super edge-magi total labeling of W T (0,, ; 3) W T (, 0, ; 3) respetively. b x x n x w y 3 y y n y d x x 3 x n x w y y 3 y n y b 3 3 n n w d b d a b x x x y y y y x x x x w w d b n 3 n 3 n y y 3 y n y d b 3 3 n x 3 w y n 3 d a Fig.. A forest W T (n, n, n 3; 3) W T (n, n, n 3; 3).

9 Super edge-magi total labeling of a tree Fig. 3. Super edge-magi total labeling of forest W T (0,, ; 3) W T (, 0, ; 3) with = 473.

10 0 K. Ali, M. Hussain, A. Razzaq Referenes. K. Ali, A. Ahmad and E. T. Baskoro, On super edge-magi total labeling of a forest of banana trees, Utilitas Math., to appear.. B. D. Aharya and S. M. Hedge, Arithmeti graphs, J. Graph Th., 4 (990), B. D. Aharya and S. M. Hedge, Strongly indexable graphs, Disrete Math., 93 (99), A. Ahmad, A. Q. Baig and M. Imran, On super edge-maginess of graphs, Utilitas Math., to appear. 5. E. T. Baskoro and A. A. G. Ngurah, On super edge-magi-total labeling, Bull. Inst. Combin. Appl., 37 (003), Z. Chen, On super edge-magi graphs, J. Combin. Math. Combin. Comput., 38 (00), H. Enomoto, A. S. Llado, T. Nakamigawa and G. Ringle, Super edge-magi graphs, SUT J. Math., 34 (980), R. M. Figueroa-Centeno, R. Ihishima and F. A. Muntaner-Batle, The plae of super edge-magi labeling among other lasses of labeling, Disrete Math., 3 (00), R. M. Figueroa-Centeno, R. Ihishima and F. A. Mantaner-Batle, On edgemagi labeling of ertain disjoint union graphs, Australas J. Combin., 3 (005), J. A. Gallian, A dynami survey of graph labeling, Eletroni J. Combin., (009).. M. Hussain, E. T. Baskoro and Slamin, On super edge-magi total labeling of banana trees, Utilitas Math., 79 (009), A. Kotzig and A. Rosa, Magi valuations of finite graphs, Canad. Math. Bull., 3(970), A. Kotzig and A. Rosa, Magi valuation of omplete graphs, Centre de Reherhes Mathematiques, Universite de Montreal, (97), CRM Slamin, M. Baa, Y. Lin, M. Miller and R. Simanjuntak, Edge-magi total labeling of wheels, fans and friendship graphs, Bull. Inst. Combin. Appl., 35(00), D. B. West, An Introdution to Graph Theory (Prentie-Hall, 996 ).