On super edge-magic total labeling of banana trees

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1 On super edge-mgic totl lbeling of bnn trees M. Hussin 1, E. T. Bskoro 2, Slmin 3 1 School of Mthemticl Sciences, GC University, 68-B New Muslim Town, Lhore, Pkistn mhmths@yhoo.com 2 Combintoril Mthemtics Reserch Division, Fculty of Mthemtics nd Nturl Sciences, Institut Teknologi Bndung Jl. Gnes 10 Bndung 40132, Indonesi ebskoro@mth.itb.c.id 3 Mthemtics Eduction Study Progrm, Universits Jember Jl. Klimntn 37 Jember, Indonesi slmin@unej.c.id Abstrct. Let G 1, G 2,..., G n be fmily of disjoint strs. The tree obtined by joining new vertex to one pendnt vertex of ech str G i is clled bnn tree. In this pper we determine the super edge-mgic totl lbelings of the bnn trees tht hve not been covered by the previous results [15]. Keywords : Super edge-mgic totl lbeling, bnn tree. 1 Introduction All grphs in this pper re finite, simple nd undirected. The grph G hs the vertex-set V (G) nd edge-set E(G). A generl reference for grphtheoretic ides cn be seen in [16]. A lbeling (or vlution) of grph is mp tht crries grph elements to numbers (usully to positive or non-negtive integers). In this pper the domin will usully be the set of ll vertices nd edges nd such lbelings re clled totl lbelings. Some lbelings use the vertex-set only, or the edge-set only, nd we shll cll them vertex-lbelings nd edge-lbelings respectively. Other domins re possible. The most complete recent survey of grph lbelings cn be seen in [10]. There re mny types of grph lbelings, for

2 2 M. Hussin, E.T. Bskoro, Slmin exmple hrmonius, cordil, grceful nd ntimgic. In this pper, we focus on one type of lbeling clled edge-mgic totl lbeling. Definition 1. An edge-mgic totl lbeling of grph G is one-to-one mp λ from V (G) E(G) onto the integers {1, 2,, V (G) E(G) } with the property tht, there is n integer constnt h such tht λ(x) + λ(xy) + λ(y) = h for ny xy E(G). It will be convenient to cll λ(x) + λ(xy) + λ(y) the edge sum of xy, nd h the mgic constnt of G. Definition 2. An edge-mgic totl lbeling λ of grph G is clled super edge-mgic totl lbeling if λ(v (G)) = {1, 2,, V (G }. The subject of edge-mgic totl lbeling of grphs hs its origin in the work of Kotzig nd Ros [11, 12], on wht they clled mgic vlutions of grphs. The notion of super edge-mgic totl lbeling ws introduced by Enomoto et l. in [5] s super edge-mgic lbeling. A number of clssifiction studies on edge-mgic totl grphs hs been intensively investigted. A prt of these studies results include Every cycle C n is super edge-mgic totl if nd only if n is odd [5]. K m,n is super edge-mgic totl if nd only if m = 1 or n = 1 [5]. K n is super edge-mgic totl if nd only if n = 1, 2, or 3 [5]. nk 2 is super edge-mgic totl if nd only if n is odd [4]. 2P n is super edge-mgic totl if nd only if n is not 2 or 3 [8]. The friendship grph consisting of n tringles is super edge-mgic totl if nd only if n is 3, 4, 5 or 7 [14]. np 3 is super edge-mgic totl for n 4 nd n even [3]. The fn F n is super edge-mgic totl if nd only if 1 n 6 [7]. However, Enomoto et l. [5] conjectured tht every tree dmits super edge-mgic totl lbeling. In the effort of ttcking this conjecture, mny uthors hve considered super edge-mgic totl lbeling for some prticulr clsses of trees for exmple [1, 2, 6, 9, 15, 17]. Lee nd Shh [13] hve verified this conjecture for trees on t most 17 vertices with computer help. Erlier, in [11] Kotzig nd Ros proved tht every cterpillr is super edge-mgic totl. However, this conjecture still remins open. The super edge-mgic totl lbeling of other clss of trees, such s bnn tree hs prtilly been discovered. The definition of bnn tree is given below. Definition 3. Let K 1,n1, K 1,n2,..., K 1,nk be fmily of disjoint strs with the vertex-sets V (K 1,ni ) = {c i, i1,..., ini } nd deg(c i ) = n i, 1 i k. A bnn tree BT (n 1, n 2,..., n k ) is tree obtined by dding new vertex nd joining it to 11, 21,..., k1.

3 Super edge-mgic totl lbeling of bnn tree n 13 2n kn k3 k2 k1 1 2 k c c 1 2 k c Fig. 1. A bnn tree BT (n 1, n 2,..., n k ). Swminthn nd Jeynthi [15] proved some results on super edgemgic totl lbeling of bnn trees. These results re s follows: BT (n 1, n 2,..., n k ), n j j, 2 j k dmits super edge-mgic totl lbeling, where n 1 is ny positive integer. In prticulr, if n 1 = n 2 =... = n k = n k, then the smllest mgic constnt of BT (n 1, n 2,..., n k ) is 2nk + 3k + 4. BT (n, 1, 1)), where n is ny positive integer, dmits super edge-mgic totl lbeling with the smllest mgic constnt 2n+16. In this pper we present super edge-mgic totl lbelings of bnn trees with some conditions tht hve not been considered in [15], nmely: BT (n 1, n 2,..., n k ), when n 1 = n 2 =... = n k = n, k 2 n k 1, nd BT (n 1, n 2,..., n k ), when n 1 > n 2 >... > n k > 1. We lso consider disjoint union of bnn trees, nmely: For m 2, n 2m, G = mbt (n, n), nd H = 2BT (n 1, n 2,..., n k ), when n 1 = n 2 =... = n k = n; n 2k, k 3.

4 4 M. Hussin, E.T. Bskoro, Slmin 2 Min Results Before giving our min results, let us consider the following lemm found in [7] tht gives necessry nd sufficient condition for grph to be super edge-mgic totl. Lemm 1. A grph G with v vertices nd e edges is super edge-mgic totl if nd only if there exists bijective function f : V (G) {1, 2,, v} such tht the set S = {f(x)+f(y) xy E(G)} consists of e consecutive integers. In such cse, f extends to super edge-mgic totl lbeling of G with mgic constnt h = v + e + s, where s = min(s) nd S = {f(x) + f(y) xy E(G)} = {h (v + 1), h (v + 2),, h (v + e)}. For Theorems 1 nd 2 let us denote the vertex nd edge sets of BT (n 1, n 2,..., n k ) s follows: V = {} {c i 1 i k} { ij 1 i k; 1 j n}, E = { i1 1 i k} {c i ij 1 i k; 1 j n}. Theorem 1. G = BT (n 1, n 2,..., n k ) dmits super edge-mgic totl lbeling if n 1 = n 2 =... = n k = n, k 2 n k 1. Proof. Consider G = BT (n 1, n 2,..., n k ), when n 1 = n 2 =... = n k = n. So v = V (G) = k(n + 1) + 1 nd e = E(G) = v 1. Now, construct lbeling λ : V E {1, 2,..., v + e} s follows: λ() = (n + 1)k + 1 k 2 nk + i, for 1 i k 2 λ(c i ) = nk i, for k 2 < i k. (n + 1)i k 2, for i k 2 λ( i1 ) = (n + 1)i n k 2, for i k The remining leves, ij where 1 i k nd 2 j n, cn be considered s the sets of vertices in which ech set mp to the set of integers ccording to the following formul.

5 Super edge-mgic totl lbeling of bnn tree 5 λ({ ij 2 j n}) = {(i 1)n + 1, (i 1)n + 2,, (i 1)n + n} \{λ( i1 )}. The set of ll edge-sums generted by the bove formul forms consecutive integer sequence nk + 2, nk + 3,, nk + (n + 1)k + 1. Therefore, by Lemm 1 λ cn be extended to super edge-mgic totl lbeling nd we obtin the mgic constnt h = v +e+s = k(n+1)+1+k(n+1)+nk +2 = 3nk + 2k + 3. Theorem 2. G = BT (n 1, n 2,..., n k ) dmits super edge-mgic totl lbeling if n 1 > n 2 >... > n k > 1. Proof. For v = V (G) = k k i=1 n i nd e = E(G) = v 1 we define the lbeling λ : V E {1, 2,..., v + e} s follows: λ(c i ) = λ() = v 2 { (v k 1) + i, for 1 i k 2 (v k) + i, for k 1 i k. λ( i1 ) = i l=1 n l, i + 2 k + i l=1 n l, for 1 i k 2 (n i + 1) + i+1 l=1 n l, for i = k 1 for i = k Agin, the remining leves ij, where 1 i k nd 2 j n i, cn be seen s the sets of vertices in which ech set mp to the set of integers ccording to the following formul. λ({ ij 2 j n}) = {α, α 1,, α (n i 1)} \ {λ( i1 )}, where α = i l=1 n l. The set of ll edge-sums generted by the bove formul forms consecutive integer sequence v + 1 k, v + 2 k,...,v + e k.therefore by Lemm 1 λ cn be extended to super edge-mgic totl lbeling nd we obtin the mgic constnt h = v +e+s = v +v 1+v +1 k = 3v k. Theorem 3. For m 2, n 2m, G = mbt (n, n) dmits super edgemgic totl lbeling.

6 6 M. Hussin, E.T. Bskoro, Slmin Proof. Let us denote the vertices nd edges of G, s follows: V (G) = { i 1 i m} { ij 1 i m, 1 j 2} { k ij 1 i m ; 1 j 2 ; 1 k n}, E(G) = { i 1 ij 1 i m, 1 j 2} { ij k ij 1 i m ; 1 j 2, 1 k n}. So v = V (G) = m(2n + 3) nd e = E(G) = 2m(n + 1). Now, construct lbeling λ : V E {1, 2,..., v + e} s follows: λ( i ) = v m + i, 1 i m, λ( ij ) = λ( i ) + i + j (2m + 2), 1 i m, 1 j 2, λ( 1 i1 ) = (n 2m + 1) + (2n + 1)(i 1), 1 i m, λ( 1 i2 ) = (2n 2m + 2) + (2n + 1)(i 1), 1 i m. Similr to bove the remining leves k ij, where 2 k n, 1 i m nd j = 1, 2, re lbelled ccording to the following formul. nd λ({ k i1 2 k n}) = {β 1 + 1, β 1 + 2,, β 1 + n} \ {λ( 1 i1 )}, λ({ k i2 2 k n}) = {β 2 + 1, β 2 + 2,, β 2 + n} \ {λ( 1 i2 )}, where β 1 = (2i 2)n nd β 2 = (2i 1)n. The set of ll edge-sums generted by the bove formul forms consecutive integer sequence 2nm+2, 2nm+3,, 2nm+2m(n+1)+1. Therefore by Lemm 1 λ cn be extended to super edge-mgic totl lbeling nd we obtin the mgic constnt h = v+e+s = m(2n+3)+2m(n+1)+2nm+2 = 6nm + 5m + 2. Exmple 1. Super edge-mgic totl lbeling of 3BT (6, 6) is given in Figure 2. Theorem 4. G = 2BT (n 1, n 2,..., n k ) dmits super edge-mgic totl lbeling if n 1 = n 2 =... = n k = n; n 2k, k 3. Proof. Let us denote the vertices nd edges of G, s follows: V (G) = { i 1 i 2} { ij 1 i 2, 1 j k} { l ij 1 i 2, 1 j k, 1 l n},

7 Super edge-mgic totl lbeling of bnn tree Fig. 2. A super edge-mgic totl lbeling of 3BT (6, 6). E(G) = { i 1 ij 1 i 2, 1 j k} { ij l ij, 1 i 2, 1 j k, 1 l n}. If v= V (G) nd e= E(G) then v = 2k(n + 1) + 2, e = v 2. Now, construct lbeling λ : V E {1, 2,..., v + e} s follows: λ( i ) = v 2 + i, 1 i 2, λ( 1j ) = λ( 1 ) 2k + (j 1), 1 j k, λ( 2j ) = λ( 2 ) k 2 + j, 1 j k, λ( 1 1j ) = (n + 1)j 2k, λ( 1 2j ) = k(n 1) + (n + 1)j 1. Similr to bove, the remining leves re lbelled ccording to the following formul. For 1 j k: nd λ({ l 1j 2 l n}) = {γ + 1, γ + 2,, γ + n} \ {λ(1 1j )}, λ({ l 2j 2 l n}) = {kn + γ + 1, kn + γ + 2,, kn + γ + n} \ {λ(1 2j )}, where γ = (j 1)n. The set of ll edge-sums generted by the bove formul forms consecutive integer sequence 2nk + 2, 2nk + 3,, 2nk + 2k(n + 1) + 1. Therefore

8 8 M. Hussin, E.T. Bskoro, Slmin by Lemm 1 λ cn be extended to super edge-mgic totl lbeling nd we obtin the mgic constnt h = v+e+s = 2k(n+1)+2+2k(n+1)+2nk+2 = 6nk + 4k + 4. Exmple 2. Super edge-mgic totl lbeling of 2BT (8, 8, 8, 8) is given in Figure Fig. 3. A super edge-mgic totl lbeling of 2BT (8, 8, 8, 8). References 1. E. T. Bskoro nd Y. Cholily Expnding super edge-mgic grphs, Proc. ITB Sins nd Tek. 36:2 (2004), M. Bc, Y. Lin, nd F. A. Muntner-Btle, Super edge-ntimgic lbeling of pth like-trees, Utilits Mth., to pper. 3. E. T. Bskoro nd A. A. G. Ngurh, On super edge-mgic-totl lbeling, Bull. Inst. Combin. Appl., 37 (2003) Z. Chen, On super edge-mgic grphs, J. Combin. Mth. Combin. Comput. 38 (2001), H. Enomoto, A. S. Lldo, T. Nkmigw, nd G. Ringle, Super edge-mgic grphs, SUT J. Mth., 34 (1980), Y. Fukuchi, A recursive theorem for super edge-mgic lbeling of trees, SUT J. Mth., 36(2000) R. M. Figuero-Centeno, R. Ichishim nd F. A. Muntner-Btle, The plce of super edge-mgic lbeling mong other clsses of lbeling, Discrete Mth. 231 (2001), R. M. Figuero-Centeno, R. Ichishim, F. A. Mntner-Btle, On edge-mgic lbeling of certin disjoint union grphs, Austrls. J. Combin. 32 (2005),

9 Super edge-mgic totl lbeling of bnn tree 9 9. R. M. Figuero-Centeno, R. Ichishim, nd F. A. Muntner-Btle, On super edge-mgic grphs, Ars Combin., 64 (2002) J. A. Gllin, A dynmic survey of grph lbeling, J. Combin. (2007), #DS A. Kotzig, nd A. Ros, Mgic vlutions of finite grphs, Cnd. Mth. Bull., 13(1970), A. Kotzig, nd A. Ros, Mgic vlution of complete grphs, Centre de Recherches Mthemtiques, Universite de Montrel, (1972), CRM S. M. Lee, nd Q. X. Shh, All trees with t most 17 vertices re super edgemgic,16th MCCCC Conference, Crbondle, University Southern Illinois, Nov Slmin, M. Bc, Y. Lin, M. Miller nd R. Simnjuntk, Edge-mgic totl lbeling of wheels, fns nd friendship grphs, Bull. Inst. Combin. Appl. 35(2002), V. Swminthn nd P. Jeynthi, Super edge-mgic strength of fire crckers, bnn trees nd unicyclic grphs, Discrete Mth., 306 (2006) D. B. West, An Introduction to Grph Theory (Prentice-Hll, 1996 ). 17. W. D. Wllis, E. T. Bskoro, M. Miller nd Slmin, Edge-mgic totl lbelings, Austrls. J. Combin., 22(2000)