On super edge-magic total labeling of banana trees

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 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.

Super edge-mgic totl lbeling of bnn tree 3 12 11 1n 13 2n 23 22 21 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 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 + 1 + 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 2 + 1. 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.

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 + 1 + 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 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},

Super edge-mgic totl lbeling of bnn tree 7 2 3 4 5 6 1 9 10 11 12 7 8 15 16 17 18 13 14 22 23 24 19 20 21 28 29 30 25 26 27 35 36 31 32 33 34 37 38 39 40 41 42 43 44 45 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 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 3. 4 5 6 7 8 1 19 36 28 45 54 3 13 14 15 16 9 23 24 17 25 18 26 33 32 34 10 22 27 35 39 40 41 42 43 44 50 51 52 53 48 49 58 59 60 61 62 31 63 12 21 47 30 56 38 57 11 20 46 55 29 2 37 64 65 66 67 68 69 70 71 72 73 74 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), 117-125. 2. 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) 82-87. 4. Z. Chen, On super edge-mgic grphs, J. Combin. Mth. Combin. Comput. 38 (2001), 55-64. 5. H. Enomoto, A. S. Lldo, T. Nkmigw, nd G. Ringle, Super edge-mgic grphs, SUT J. Mth., 34 (1980), 105-109. 6. Y. Fukuchi, A recursive theorem for super edge-mgic lbeling of trees, SUT J. Mth., 36(2000) 279-285. 7. 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), 153-168. 8. R. M. Figuero-Centeno, R. Ichishim, F. A. Mntner-Btle, On edge-mgic lbeling of certin disjoint union grphs, Austrls. J. Combin. 32 (2005), 225-242.

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