Exclusive Sum Labeling of Graphs: A Survey

Transcription

1 AKCE J. Graphs. Combn., 6, No. 1 (2009), pp Exclusve Sum Labelng of Graphs: A Survey Joe Ryan School of Electrcal Engneerng and Computer Scence The Unversty of Newcastle Callaghan, NSW 2308 Australa e-mal: joe.ryan@newcastle.edu.au Abstract All sum graphs are dsconnected. In order for a connected graph to bear a sum labelng, the graph s consdered n conjuncton wth a number of solated vertces, the labels of whch complete the sum labelng for the dsjont unon. The smallest number of solated vertces that must be added to a graph H to acheve a sum graph s called the sum number of H; t s denoted by σ(h). A sum labelng whch realzes H K σ(g) as a sum graph s called an optmal sum labelng of H. In ths paper we survey a new type of labelng based on summaton, the exclusve sum labelng. A sum labelng L s called exclusve sum labelng wth respect to a subgraph H of G f L s a sum labelng of G where H contans no workng vertex. The exclusve sum number ɛ(h) of a graph H s the smallest number r such that there exsts an exclusve sum labelng L whch realzes H K r as a sum graph. A labelng L s an optmal exclusve sum labelng of a graph H f L s a sum labelng of H K ɛ(h) and H contans no workng vertex. Keywords: sum graphs, sum number, optmal sum labelng, exclusve sum labelng, exclusve sum number, optmal exclusve sum labelng Mathematcs Subject Classfcaton: 05C Introducton All graphs we consder here are fnte, smple and undrected. For general terms used n graph theory, please refer to [2]. A sum labelng λ of a graph G s a mappng of the vertces of G nto dstnct postve ntegers such that for u, v V(G), uv E(G) f and only f the sum of the labels assgned to u and v equals the label of a vertex w of G. In such a case w s called a workng vertex. A graph whch has a sum labelng s called a sum graph. Sum graphs were orgnally proposed by Harary [6] and later extended to nclude all ntegers [7]. Sum graphs cannot be connected graphs snce an edge from the vertex wth the largest label would necesstate a vertex wth a larger label. Graphs whch are not sum graphs

2 114 Exclusve Sum Labelng of Graphs: A Survey can be made to support a sum labelng by consderng the graph n conjuncton wth a number of solated vertces whch can bear the labels requred by the graph. The fewest number of the addtonal solates requred by the graph to support a sum labelng s called the sum number of the graph; t s denoted by σ(g). Every edge adjacent to the vertex bearng the largest label requres an solated vertex to wtness the edge. Consequently a lower bound for the number of solates requred for a graph to support a sum labelng s δ(g) - the smallest degree of G. Any graph for whch σ(g) =δ(g) s known as a δ-optmal summable. A sum labelng of a graph G K r for some postve nteger r s sad to be exclusve wth respect to G f all of ts workng vertces are n K r. Every graph can be made to support an exclusve sum labelng, by addng a requred number of solates. The least possble number of solates that need to be added to a graph G to obtan an exclusve sum labelng s called the exclusve sum number of the graph G, denoted by ɛ(g). Observaton 1. ɛ(g) (G). Let (G) be the maxmum degree of the vertces of a graph G. Then In case ɛ(g) = (G), the graph G s sad to be a -optmum summable graph. Clearly, every exclusve sum graph s a sum graph but not vce versa and so the exclusve sum number s always greater than or equal to the sum number, that s, Observaton 2. For any graph G, ɛ(g) σ(g). There are several graphs for whch σ(g) = ɛ(g) and the optmum sum labelng s exclusve. Examples of such graphs nclude complete graphs (K n ) [1], cocktal party graphs (regular complete multpartte graphs (H m,n )) [12] and odd wheels (W n,n odd) [9, 11]. Of these examples only the odd wheels are -optmum summable. An nterestng result of Gould and Rödl [5] s that there exst famles of sum graphs whose sum number s quadratc n the order of the graph. Despte the work of Nagamoch et al. [13] showng that almost all graphs have superlnear sum number, no famles of graphs have been found to exhbt ths feature. In lght of Observaton 2, these results must hold for exclusve sum graphs and whle these may prove easer to fnd, to date none have been dscovered. 2. Lnear Transformatons It s clear from the nature of sum graphs that f L s a sum labelng of a graph G then so s αl, where α s any postve nteger. The case nvolvng addton s not qute so clear. For L a sum labelng, let L + β be the labelng formed by addng β to every label. To mantan the sum labelng propertes we need to add 2β to workng vertces, or more f the workng vertex s wtnessng an edge between workng vertces. In exclusve sum graphs, adjacent vertces are always non-workng, so addng β to every vertex n the graph and 2β to the solates mantans the exclusve sum labelng propertes.

3 Joe Ryan 115 In what follows, when we speak of a lnear transformaton on a labelng, we mean transform L(u) tok 1 L(u)+k 2 for u a non-workng vertex and k 1 L(v)+2k 2 for v a workng vertex. Mller et al. showed n [17] that we can use a lnear transformaton to ensure that the labelng has partcular propertes such as a specfc mnmum value, however not all transformatons wll be sutable to mantan the sum labelng propertes as the followng example shows. Table 1 gves a (possbly non-optmal) labelng for H 4,4. The partte sets, labeled v 1 v 4 etc. are presented n rows n the top part of the table whle below are lsted (and labeled w 1 etc) the solates. One method of relabelng ths (exclusve) labelng so as to acheve a mnmum label 1 would be to add 41 to all vs and 82 to all ws (.e. set k 1 =1 and k 2 = 41). Ths transformaton would label the vertces v 5 v 9 wth 2, 4, 6, 8thus nducng edges n a partte set. v 1 v v 5 v v 9 v v 13 v w 1 w w 7 w w 14 w w 21 w Table 1: A sum numberng for H 4,4. Substtutng k 1 =2,k 2 = 83 relabels the frst three solates wth 4, 8, 12 thus nducng edges among the solates. k 1 =3,k 2 = 125 assgns labels 5, 17 to w 1 and w 3 respectvely whle v 8 s assgned 22, agan nducng edges between solates, ths tme wtnessed by a vertex from wthn the graph. The followng theorem explans the problems from the above example and provdes a strategy for choosng a transformaton that avods nducng unwanted edges. Theorem 1. If L s an exclusve sum graph labelng of a graph H n G = H K r then so s the labelng L (u) =k 1 L(u)+k 2 for u H and L (u) =k 1 L(u)+2k 2 for u K r, where k 2 s any nteger whch results only n postve dstnct values n L and k 1 s any postve nteger that does not dvde 6k 2. Proof. If w s a vertex wtnessng an edge between u and v then u + v = w and n the new labelng u + v = k 1 u + k 2 + k 1 v + k 2 = k 1 (u + v)+2k 2 = k 1 w +2k 2 = w

4 116 Exclusve Sum Labelng of Graphs: A Survey Then all edges n the orgnal graph wll stll be present under the new labelng. It remans to show that no new edges are nduced. There are four cases to consder. Case 1. An extra edge may be nduced wthn the graph H whenever u + v = w where u, v, w H. We need to ensure u + v w k 2 (1) k 1 Ths can be ensured by forbddng the fnal term from beng an nteger, so we choose a k 1 that does not dvde k 2. Case 2. An extra edge may be nduced between workng vertces by another workng vertex whenever u + v = w wth u, v, w K r. Here we need to ensure u + v w 2k 2 k 1 (2) Case 3. An extra edge may be nduced between workng vertces by a non-workng vertex whenever u + v = w wth u, v K r and w H. To avod ths case we need u + v w 3k 2 k 1 (3) Case 4. Extra edges may be nduced between H and the workng vertces n two dfferent ways. Frst when w K r and secondly when w H. The former stuaton s addressed by (1) and the latter by (2). All three nequaltes may be realsed by ensurng that k 1 does not dvde 6k 2. v 1 v v 5 v v 9 v v 13 v w 1 w w 7 w w 14 w w 21 w Table 2: A mnmal sum numberng for H 4,4. Therefore, for the lnear transformaton to be sutable we need to answer the followng queston.

5 Joe Ryan 117 Gven postve ntegers P and Q, can we always fnd ntegers s and t where s>0and does not dvde 6t such that P = sq + t? Ths can be easly answered n the affrmatve by choosng an s 2, 3, 6thatdoesnot dvde P and t = P sq. In the prevous example we can thus choose k 1 =5,k 2 = 209 asshownntable. 3. Exclusve Sum Labelng of Complete Bpartte Graphs Hartsfeld and Smyth [9] showed that the sum number of a complete bpartte graph K p,q for q p 2sequalto. In 2001 He et al. [10] showed that ths result s 3p+q 3 2 only realsed for a lmted range of p and q. The correct sum number for K m,n was gven ndependently by three sets of authors [18],[10], [14], all publshed n the same ssue of Dscrete Mathematcs n However, n the case of exclusve sum number, the followng lemma whch s modfed from the Hartsfeld s et al. paper [9] s true n general and provdes a lower bound on the exclusve sum number for the graph K p,q. Lemma 1. For q 2 and p 2,ɛ(K p,q ) p + q 1. Proof. Let L be any exclusve sum labelng of a complete bpartte graph K p,q,q 2, p 2. Let P and Q be the two partte sets, where P = p 2, Q = q 2. Suppose that the labels of P = {x 1,x 2,..., x p } under L are arranged nto an ascendng sequence, so that x j <x j+1, 1 j p 1. Smlarly, arrange the labels of Q = {y 1,y 2,..., y q } nto an ascendng sequence. Observe that each of the followng sums s dstnct x 1 + y 1 <x 2 + y 1 <... < x p + y 1 <x p + y 2 <... < x p + y q. Snce there are exactly p + q 1 dstnct sums, t follows that at least p + q 1 solated vertces are requred to label the graph exclusvely. Next let k> max {2p 2,p+ q 2} and suppose that L labels the vertces of P and Q as follows. P = {1+4 0 p 1} Q = {1+4j k j k + q 1} Let R be the set of solated vertces whch are labeled by {(1 + 4)+(1+4k) 0 p 2} {(1 + 4(p 1)) + (1 + 4j) k j k + q 1}. It s clear that R = p + q 1. Note that the labels used for P and Q are 1 (mod 4) and the labels used for the solated vertces R are the sums of two numbers of 1 (mod 4), that s, 2 (mod 4). Therefore, K p,q contans no workng vertex.

6 118 Exclusve Sum Labelng of Graphs: A Survey The sum of any two numbers from P or Q cannot be n R bythechoceofk. Moreover, snce numbers congruent to 3 (mod 4) and 0 (mod 4) do not occur n ths labelng, we conclude that no extra edges are nduced between the solates or between the graph and the solates. Therefore, we have shown that L s an exclusve sum labelng of K p,q whch realses the lower bound of ɛ(k p,q ). Hence we have the followng theorem: Theorem 2. For q 2 and p 2,ɛ(K p,q )=p + q 1. In the next secton we present a constructon of exclusve sum labelng for paths and cycles and we gve ther exclusve sum numbers. 4. Paths and Cycles 4.1. Paths Let v 1,v 2,..., v n be the vertces of the path P n. Label the vertces v wth v = x+( )n, for odd and v =2x ((/2) 1)n for even, where x>n(n 2), then v + v +1 =3x for odd and v + v +1 =3x + n for even. Thus P n has an exclusve labelng wth 2 solated vertces, v n 2 + v n 1 and v n 1 + v n. We have just proved Theorem 3. The exclusve sum number for paths, ɛ(p n )=2, for n Cycles We start ths subsecton wth the followng lemma. Lemma 2. C n. There are at least three dstnct edge labels n any exclusve sum labelng of Proof. It s obvous for n =3. Now we assume that n>3. Let w be the largest vertex on C n. Let u and v be adjacent to w, and (u, w) and (v, w) be ther correspondng edges. Wthout loss of generalty, we suppose that u<v.snce n>3, t follows that there s a vertex t and a correspondng edge (t, u). It s clear that t + u u + w and u + w v + w. Snce v>uand w>t,we see that v + w u + t. Therefore, the sums t + u, u + w and v + w are all dfferent. It s necessary to deal separately wth odd and even cycles.

7 Joe Ryan Odd Cycles Let v, for 1 n, be the vertces on the cycle C n for odd n. Suppose that we label the vertces as follows. v = where v 1 =1,v 2 = v n d 2 and { v 2 + d for odd 3 v 2 d for even 4 { 2 n d = 4 f 2 n otherwse. Now we sum each par of adjacent vertces n the two dfferent cases. () For odd we have, v + v +1 =(v 2 + d )+(v 1 d +1 ). Note that for odd, d and d +1 have the same value. Therefore, v + v +1 = v 2 + v 1 = v 1 + v 2 = 1+v n d 2 = 1+v n 2. 4 () For even, we consder the followng three cases. 1. for <2 4 and +1< 2 4, v + v +1 = (v 2 d )+(v 1 + d +1 ) = (v 2 2 )+(v 1 +2 ) 4 4 = v 2 + v 1 = v 2 + v 3 = (v n d 2 )+(v 1 + d 3 ) = v n + v 1 = v n +1

8 120 Exclusve Sum Labelng of Graphs: A Survey 2. for =2 n 4 and +1> 2 n 4, v + v +1 = (v 2 d )+(v 1 + d +1 ) = (v 2 2 )+(v ) = (v 2 2 )+(v ) for >2 4 and +1> 2 4, = v 2 + v 1 +2 = v 2 + v 3 +2 = (v n d 2 )+(v 1 + d 3 )+2 = v n + v 1 +2 = v n +3 v + v +1 = (v 2 d )+(v 1 + d +1 ) = (v )+(v ) ThssthesameasCase2.Thus = v 2 + v 1 = v 2 )+(v v + v +1 = v n +3 We see that there are three dstnct edge labels of the cycles C n for odd n, that s, 1 + v n 2 n 4,vn +1 and v n +3. Thus, n vew of Lemma 2, the constructon of an exclusve labelng for odd cycles requres three solated vertces Even Cycles Let v V (C n ), 1 n, n even. Suppose that we label the vertces as follows. { 4 3 f s odd v = 4n 4 +5 f s even Then the sum of each par of adjacent vertces s: for 1 n 1, { 4n 2 f s odd v + v +1 = 4n +6 f s even and v n + v 1 =6 In vew of Lemma 2, an optmum exclusve sum labelng of even cycles requres three solates and so we have

9 Joe Ryan 121 Theorem 4. ɛ(c n )=3, for n Trees We refer to [3] for the notons of tree, caterpllar, and shrub. However, t s worth mentonng some terms used n that paper whch wll be used n ths paper. 1. A leaf of a tree s a vertex wth degree A near-leaf s a non-leaf that has at most one neghbour whch s not a leaf. 3. An nner vertex s a vertex that has at least two neghbours whch are not non-leaf. In [3], M.N. Ellngham proved that f T s a tree of order at least 2, then σ(t )=1. Snce the ntroducton of the notons of exclusve sum labelng and exclusve sum number of graphs, t has been beng a challenge to fnd the exclusve sum number of trees. Unlke ts counterpart problem n sum number, attempts to solve ths problem so far are stll unsuccessful. However, some results have come to hand. Some trees are -optmum summable graph, but there exsts trees whch are not -optmum summable [16]. For example, caterpllars and shrubs (central n Ellngham s proof) along wth stars and double stars are -optmum summable graph. The result s cannot be generalsed and we conclude wth a tree whch s not -optmal summable. Recall that a caterpllar s a graph whch has the property that f we remove all the vertces wth degree 1 then what remans s a path. A caterpllar can have more than one longest path. Such a path s called the spne of the caterpllar. The two end ponts of a spne are called respectvely by tal and head. Other vertces at the spne are called the nternal vertces. We shall always consder a spne of a caterpllar as orented n partcular drecton from tal to head. The vertces of degree one of a caterpllar other than tal and head wll be called the feet, whch are attached to the nternal vertces of ts spne by edges called the legs of the caterpllar. Let C be a caterpllar wth (C) =d. Labelng 1 (Exclusve sum labelng of caterpllar) 1. Choose a spne of C and let P = {p 1,p 2,..., p n } be the the set of vertces of the spne. Let f = deg(p ) 2, =2, 3,..., n 1. = 0, =1,n. For 2 n 1, let B = {b j 1 j f } be the set of feet whch are attached to the nternal vertex p. It s clear that B = N(p )\(P N(p )) for 2 n 1. Let B = n 1 B be the set of all feet of C. =2

10 122 Exclusve Sum Labelng of Graphs: A Survey 2. Label the spne wth a mappng L as follows. It gves: L(p ) = 1+2( 1)(d 2) for odd, = 1+4(n /2)(d 2) for even. L(p )+L(p +1 ) = 2+(4n 4)(d 2), for odd, = 2+4n(d 2), for even. 3. Let T A = {2+(4n 4)(d 2), 2+4n(d 2)} and choose a>max(t A )), a 1(mod 4). 4. Add two solates and label wth the number from T A. 5. Add more d 2 solates T B = {t (b) =1, 2,..., (d 2)} and label wth L(t (b) )= (a + L(p 2 )) + 4( 1). 6. Let T = T A T B, and for some k=1,2 and l=1,2,...,d-2, let t (a) k Label the vertces of B as follows. L(b j )=L(t (b) j ) L(p ),j =1, 2,..., f. T A and t (b) l T B. For convenence, from now on each vertex wll be consdered as label under L. We wll use v for example nstead of L(v) for any v V (C) K d. Remark 1. For =1, 2,..., n let B = {t(b) j p j =1, 2,..., d 2}. It s obvous that B B,=1, 2,..., n. If B = n B then B B. =1 Before we prove that ths labelng s an optmal exclusve sum labelng of C. We need to consder the followng facts: Observaton max(p )=p 2, mn(p )=p 1, 2. max(t A )=p 2 + p 3, mn(t A )=p 1 + p 2 3. max(t B )=a + p 2 +4(d 3), mn(t B )=a + p 2 4. max(b )=max(t B ) mn(p )=a + p 2 +4(d 3) p 1, and mn(b )=mn(t B ) max(p )=a + p 2 p 2 = a. Lemma 3. Let p P, t (a) T A,b B and t (b) T B, where P, T A and T B as n Labelng 1, then p<t (a) <b<t (b).

11 Joe Ryan 123 Proof. There are three parts to prove, 1. We wll show that p<t (a) for all p P and for all t (a) T A. Let p P, t (a) T A,then p p 2, and ether t (a) = p 2 + p 1 or t (a) = p 2 + p 3. In both case, we have p 2 <t (a). Ths gves, p<t (a) for all p P and for all t (a) T A. 2. We wll show that t (a) <b, for all t (a) T A and for all b B. Let b B and t (a) T A then, b a>max(t A ) t (a). Therefore, b>t (a), t (a) T A, and b B. 3. We wll show that t (b) >b, for all t (b) T B and for all b B. Let t (b) T B and b B, then t (b) mn(t B )=a+p 2. max(b )=max(t B ) mn(p )=(a+p 2 )+2(d 3) p 1. a + p 2 max(b ) = (a + p 2 ) (a + p 2 )+2(d 3) p 1 = p 1 2(d 3) = 2d 2(d 3) > 0 We have, a + p 2 > max(b ), therefore t (b) > max(b) b From these three facts we have p<t (a) <b<t (b), p P, t (a) T A,b B and t (b) T B. Lemma 4. Let P, B r and B s as n Labelng 1. If r s then B r B s = Proof. x = t (b) x = t (b) j We get t (b) Suppose on the contrary, B r B s. Let x B r B s. Then p s for some t (b) T B and p r for some t (b) j T B. t (b) j = p s p r. But, t (b) t (b) j max{ t (b) u On the other hand, p s p r 4(d 2). t (b) Hence, t (b) t (b) j t (b) j = p s p r. t (b) v t (b) u,t (b) v T B } = max(t B ) mn(t B ) = 4(d 3) 4(d 3) < 4(d 2) p s p r. Ths contradcts to the fact that Therefore we must have B r B s =. As a consequence,

12 124 Exclusve Sum Labelng of Graphs: A Survey Lemma 5. If r s then B r B s =. Observaton 4. For all, =1, 2,..., n p 1(mod 4) and b j 1(mod 4) for all j =1, 2,..., f. Ths gves t 2(mod 4) t T. We are ready to proof the followng theorem. Theorem 5. Let C be a caterpllar wth vertex maxmum degree (C) then ɛ(c) = (C), that s, a caterpllar s -optmum sum graph. Proof. We wll show that Labelng 1 gves an exclusve sum labelng to the caterpllar C. By Lemma 3, the labelng s surely a bjecton from V (C T ) onto dstnct postve nteger numbers n P B T A T B. Moreover,tsclearthatf{u, v} E(C) then u+v T. We need to prove that, there s no extra edge needed,that s, f {u, v} / E(C T ) then u + v/ V (C T )=P B T A T B. 1. Let x B and y B,{x, y} / E(C). Obvously x + y/ B and x + y/ P. We wll show that x + y/ T. x + y>t, t T A due to Lemma 3. Notce that a>p 2 + p 3. Therefore x + y/ T B. x + y 2a > (a + p 2 )+p 3 > (a + p 2 )+1+4(d 2) > (a + p 2 )+4(d 3) = max(t B ). 2. Let x B, y P and {x, y} / E(C). It s obvous that x + y/ P and x + y/ B. We wll show that x + y/ T. Let x = b j and y = p k where k. Due to Lemma 3, b j >t, t T A. So b j + p k / T A b j = t p for some t T B. If b j +p k T B, then b j +p k = t for some t T B. We get b j = t p k Ths says that b j B B k, whch s contradcts to Lemma Let {p,p j } / E(C).Clearly p +p j / P and p +p j / B. We wll show that p +p j / T. If p +p j = t for some t T A, then p = t p j. Ths gves p = p j 1 or p = p j+1. Ths of course contradcts to the fact that p,p j / E(C). If p + p j = t for some t T B, then p = t p j B j. Ths contradct to Lemma 1 that p<b p, b.

13 Joe Ryan 125 By Observaton 2, there s no possblty for the occurrence of any unwanted edge. The above labelng, s an exclusve labelng of C wth the addton of (C) solates. 6. Concluson The major unsolved problems n ths area nclude the analog to Gould and Rödl s result (as mentoned n the Introducton) and the classfcaton of -optmal trees. There are, however many avenues for further research n exclusve sum graphs. The condtons for ensurng a lnear transformaton s an exclusve sum labelng are too strong and somethng weaker than k 1 not beng a factor of 6k 2 should suffce. In addton to these, there are many graphs wth known sum number whose exclusve sum number has not been nvestgated. Tuga and Mller [15] have developed a technque for fndng optmal exclusve sum labelngs for certan graphs wth radus 1 ncludng fans (also known as shells), multfans (multshells) and frendshp graphs. Another nterestng area mght be the study of graphs that are δ-optmal and -optmal. The only members of ths class to date are caterpllars (and subsets ncludng paths and stars). In [4] Fernau et al. showed that generalsed frendshp graphs are δ-optmal. Gven the result of Tuga and Mller n [15], the generalsed frendshp graph mght be the frst non caterpllar nto the ranks of the double delta optmal graphs. References [1] D. Bergstrand, F. Harary, K. Hodges, G. Jennngs, L. Kuklnsk and J. Wener, The sum number of a complete graph, Bull. Malaysan Math. Soc., 12 (1989), [2] G. Chartrand and L. Lesnak, Graphs and Dgraphs, 3rd Edton, Chapman and Hall [3] M. Ellngham, Sum graphs from trees, Ars Combn., 35(1993), [4] H. Fernau, J. Ryan and K. A. Sugeng, A sum labelng for the generalsed frendshp graph, Dscrete Math., 308, (5 6) (2008), [5] R. Gould and V. Rödl, Bounds on the number of solated vertces n sum graphs, Graph Theory, Combnatorcs and Applcatons (edted by Y. Alev, G. Chartrand, O.R. Oellermann and A.J. Schwenk) John Wley and Sons (1991), [6] F. Harary, Sum graphs and dfference graphs, Congressus Numerantum, 72 (1990), [7] F. Harary, Sum graphs over all the ntegers, Dscrete Math., 124 (1994),

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