VASANTI N BHAT-NAYAK and UJWALA N DESHMUKH*

Transcription

1 Proc. ndan Acad. Sc. (Math. Sc.), Vol. 106, No. 2, May 1996, pp Prnted n nda New famles of graceful banana trees VASAN N BHA-NAYAK and UJWALA N DESHMUKH* Department of Mathematcs, Unversty of Bombay, Vdyanagar, Bombay , nda * Mthba College of Arts and Scence, Vle Parle, Bombay , nda MS receved 31 January 1994; revsed 24 July 1995 Abstract. Consder a famly of stars. ake a new vertex. Jon one end-vertex of each star to ths new vertex. he tree so obtaned s known as a banana tree. t s proved that the banana trees correspondng to the famly of stars ) (KL1,K1, 2... Kla_l,(otl)KLt, Kl,~ 1... KL.), o~>~0, ) (2KL,2KL2... 2K,t_,(~2)KLt, 2KL,... 2KL,), 0~<~<tand ) (3KL,3KL2... 3K1,.) are graceful. Keywords. rees; graceful. 1. ntroducton A graph G wth p vertces and q edges s graceful f there s an njectve map q~ from the vertex set V of G nto (0, 1... q) such that the nduced map ~from edge set E of G nto (1, 2... q) defned by ~(e) = kb(u) - ~b(v)[ where e = uv, s surjectve. Such a ~b s known as a graceful labelng of G, ~(e) s known as weght of e nduced by c~. A tree s a connected acyclc graph. For a tree q = p - 1. A well-known conjecture due to Rngel and Kotzg [3, 2] s that all trees are graceful. hs conjecture s stll unsettled. A banana tree [1] s one obtaned from a famly of stars by jonng one end-vertex of each star to a new vertex. We prove that a banana tree correspondng to the famly of stars (K~,, K1, 2... Kl,t- 1, (0c 1) Kl,t, K 1,t 1... K,,), 0 <~ ~ s graceful when ~ < t. Usng ths result we show that a banana tree obtaned from the famly of stars (2Kl,1,2K1, z... 2Kl,t- 1, (~ 2) Kn, 2Kl,t K1.,) s graceful when 0 ~< ~ < t. Chen, L and Yeh [-1] have proved that a banana tree obtaned from the famly of stars (2Kl,l,2K1, K1,.) s graceful. We gve a dfferent graceful labelng for ths banana tree and use t to establsh that a banana tree obtaned from the famly of stars (3K1,1,3K1,2...,3K1,.) s graceful. he banana tree obtaned from the famly of stars (K1,1, KL2,... K,.) s known as a standard banana tree [1] and we denote t by SB". We also establsh that a banana tree correspondng to the famly of stars (K,,K1, 2... Kl,t_l,(ot "4-1)Kt,t, Ka,t a... KL. ), ~>~ t, s graceful. However, our graceful labelng here s not amenable to combnng ths banana tree wth the standard banana tree SB" to get a gracefully labeled banana tree whch has (2K1,1,2K1, Kl,t_ 1,(ct 2)Km,2Km K1,.) as the famly of stars when ~/> t unlke the case when 0t < t. We use the followng termnology. Consder a banana tree obtaned from the famly of stars (K 1... K , Kl,x.). he new vertex to whch these stars are attached s called the apex. he end-vertces of these stars whch are joned to the apex are called 201

2 202 l/asant N Bhat-Nayak and Ujwala N Deshmukh *.o ~ * u247 ~~ ~ u,.q!! u u E --F B -F o 8 uq ~ ~ 4- ~ ~ ~ 247 d- : H V -F d--k ~ H 247 bo t"xl ~ -F o-t ~g.q Jr Jr.=_ b5 0 Z

3 New famles of graceful banana trees 203 lnk vertces. he vertex of Kl,x, of degree x s called ts central vertex. Note that all vertces of K.~, other than the central vertex and the lnk vertex are end-vertces of the banana tree. 2. Graceful labelng of SB"(t, ~e), ~ < t he banana tree obtaned from the famly of stars (K 1.1' K K la- 1, (~ 1) K.,, K1,,1... K t,,) s denoted by SB"(t, ~), where ~ < t. Clearly, SB"(t, ~) s a tree wth q = (n(n 1)/2) n 0ct 0~ edges. he star K1., s repreated ~ 1 tmes n the famly. We denote them by K 1 r,'*~ ~*-, ~c*,, t' ~1, t~ "~1, t'" " "' ~ 1,," able 1 ndcates the labelng of SB"(t, ~), 9 < t and also the nduced weghts of the q edges and clearly ponts out that ths labelng s graceful. Label 0 s gven to the apex. Fgure 1 shows a graceful labelng of 8B7(5, 3) gven as per table 1. For convenence, the apex and the lnk edges are not drawn. he topmost vertex of each star s ts lnk vertex. 3. Graceful labelng of 2 - SB'(t, a~), a~ < t he banana tree correspondng to the famly of stars (2K1.1,2K1, K1, t-1, (~ 2)Kl.,,2K1., x... 2K1.,) s denoted by 2-SB"(t, ct) where ~ < t. he banana tree 2 - SB"(t, ~) can be thought of as the one obtaned from the standard banana tree SB" and the banana tree SB"(t, ~), ~ < t, by dentfyng ther apex vertces. Chen, L and Yeh [ 1] have gven a graceful (n fact nterlaced) labelng of S B" whch s gven n table 2. Fgure 2 gves a graceful labelng of SB 7. Now we gve an algorthm for obtanng a graceful labelng of 2- SB"(t, ~), ~ < t, us~ng tle graceful labelngs of SB"(t, ~) and SB" as gven n tables 1 and 2. t } ~, t. 1 t.o Fgure 1. Graceful labelng of SB 7(5, 3) : Fgure 2. Graceful labelng of SB 7.

4 204 l/asant N Bhat-Nayak and Ujwala N Deshmukh -- o H, ~ q-!!!! 1 fl ~o"s ~ ~ o ~ -F Jr.t=.2.B ~ 0 --g.-g 4-.=_..3,.r 0 Z

5 SB" Algorthm New famles of graceful banana trees 205 Step 1. dentfy the apex vertces of SB" and SB" (t, e), e < t. Gve label n to the apex of 2 - SB"(t, e) thus obtaned. Step 2. Add n to each label of SB"(t,e) except to the apex whch s now the apex of 2 - SB"(t, e) and s covered by step 1. Step 3. For all the central vertces of SB ~ change label x to n - x. Step 4. For all other vertces of SB" change label x to n z 4n e(t 1) 1 - x. Remark. Note that q(2 - SB"(t, e)) = n 2 3n e(t 1). Also note that x >~ n 1. able 3 gves the edge weghts of 2 - SB"(t, ~), ~ < t, when ths algorthm s executed. able 3 does not nclude the edge weghts of edges from SB"(t, e) part of 2 - SB"(t, e) snce they are exactly the same as those of SB"(t, ~) and these are already lsted n table 1. hese account for weghts 1 to ((n 2 3n)/2) e(t 1) = q(sb"(t, e)). Let f12 = (( n2 3n)/2) ~(t 1) = q(sb"(t,~)). he edge weghts (f12 1) to (q(2 - SB"(t, e) = fl2((n 2 3n)/2) are covered by the edges of SB * part of 2 - SB"(t, ~), Fgure3 gves a graceful labelng of 2-SB7(5,3) obtaned by our technque. Fgure 3 s to be read wth fgures 1 and 2. As before, the apex and lnk edges are not shown. he apex gets label Graceful labelng of 3 - he banana tree correspondng to the famly of stars (3K1, r,3k1, K1,,) s denoted by 3 - SB". he banana tree 3 - SB" dan be thought of as the one obtaned from SB ~ and 2- SB ~ by dentfyng ther apex vertces. Here 2- SB ~ denotes the banana tree correspondng to the famly of stars (2K1, 1,2K~, 2...,2K1.,). Chen, L and Yeh [1] have gven a graceful (n fact nterlaced) labelng of 2 - SB ". nterlaced labelng s a specalzed graceful labelng. However, ther graceful labelng of 2 - SB" and ther technque of combnng nterlaced banana trees to get a larger nterlaced, and hence graceful, banana tree does not gve a graceful labelng of 3 - SB ". he reason s, n ther nterlaced labelng of 2 - SB ~, the apex receves the label n (and not 0) /..6 &O g 5/, /-,8 47 t, 5 4,:, 1,3,'.~ /,1 35 3/, '5 2/, lg , 6"~ /* B g Fgure 3. Graceful labelng of 2 - SB7(5, 3). he apex gets label 7.

6 ,~06 Vasant N Bhat-Nayak and Ujwala N Deshmukh o E 2~2g E o j ~ 4-! -~ ~..~. ] ~! ~' 1 ol?.l ol 1% f f '2. E o e-, e~ ~ ~Y ~E ~R ~ r {D,4??-~7? e~ ~2.E Q Z Z

7 New famles of graceful banana trees E ~! ~ ~ l ~'o ~".2 ~'o ~" t- t'q f ~ ~ o ~ M m o Z

8 208 Vasant N Bhat-Nayak and Ujwala N Deshmukh? 1 eb!! ~ 8, 1 ~ ~ o 9. -t- o k "2. r- {,? : {} Z

9 New famles of graceful banana trees 209 ~'~ ~ ~- x. ~,,.~o~,,.~ 1 o t -- :! -- "~ -- -t- -- ='- / -t- ~,o = ~ ~ _~ -t- ~ ~.m r~ Z._~

10 210 Vasant N Bhat-Nayak and Ujwala N Deshmukh 7~ 7 ~ 7!!? L ~ r ~o~ ~ Q!! 7 tr ~ ~ 7 7 r,~,~ '~ Lu ~ ~o ~ ~ 7~ ~7 r.) o z

11 New famles of graceful banana trees 211 Here we gve a dfferent graceful (but not nterlaced) labelng of 2 - SB n n whch the apex gets the label 0. hs new labelng depends on the party of n. We consder 2 - SB" as two copes of SB" wth the same apex. able 4.1 gves a labelng of one copy of SB", table 4.2 gves the other copy of SB * when n s even and table 4.3 the other copy of SB ~ when n s odd. able 4.2 has, n fact, two parts. We have called them table 4.2 (even) and table 4.2 (odd). able 4.2 (even) ndcates the labelng for even szed stars nvolved n SB n, that s, for K 1,2, K 1, 4... able 4.2 (odd) ndcates the labelng for odd szed stars nvolved n SB ", that s, for K,,KLa... Smlarly we have table 4.3 (even) and table 4.3 (odd). Fgure 4.1 shows a graceful labelng of 2 - SB s whle fgure 4.2 shows a gracefullabelng of 2 - SB 7. We have ncluded a graceful labelng of SB s to be used for that of 2 - SB s. Now we gve an algorthm to gracefully label 3 - SB" where 3 - SB ~ s obtaned from SB" and 2 - SB ~ by dentfyng ther apex vertces. Algorthm Step 1. Do the graceful labelng of SB" part of 3 - SB" as per table 2. Step 2. Do the graceful labelng of 2 - SB" part of 3 - SB" as per tables 4.1, 4.2 and 4.3. Step 3. dentfy the apex of SB n wth that of 2 - SB" and gve label n to t. Step 4. Add n to each label of 2 - SB n (except to the apex whch s covered by step 3). 4/-, Fgure 4.1. Graceful labelng of SB s " BO / ~9 18 1~" ~2 11 ~ /.3 Fgure 4.1. (contnued). Graceful labelng of 2 - SB s.

12 212 Vasant N Bhat-Nayak and Ujwala N Deshmukh t 222L , L9 48 /-,7 1, l* ~ /, Fgure 4.2. Graceful labelng of 2 - SB'. Step 5. Change label x of SB" to n - x for all the central vertces of SB". Step 6. For all other vertces of SB" change label x to 3 ((n 2 3n)/2) n 1 - x. Remarks 1. Note that 3((n 2 3n)/2) = q(sb") q(2 - SB"). Also note that x >/n Steps 3, 4, 5 and 6 of Algorthm are same as steps 1, 2, 3 and 4 of Algorthm wth q(sb"(t, ~)) replaced by q(2- SB"). n other words, technques of the algorthms are same. able 5 gves the edge weghts of 3 - SB" when ths algorthm s executed. hs table does not nclude the edge weghts of 2 - SB" part of 3 - SB ~ snce they are exactly the same as those of 2 - SB" and are already lsted n tables 4.1 and 4.2. Fgures 5(even) and 5(odd) gve graceful labelngs of 3 - SB 8 and 3 - SB 7 respectvely. hese fgures are to be read along wth fgures 4.1, 2 and 4.2 respectvely. As before, for convenence, the apex and the lnk edges are not drawn and the top-most vertex of each star s ts lnk vertex. 5. Graceful labelng of SW(t, ~) wth arbtrary t and ~e n w 2 we gave a graceful labelng of SB"(t, c 0 when c~ < t. Here we gve a graceful labelng of SB"(t, ~) wthout any restrcton on c~ and t. Let B(eK1, t) denote the banana tree whose famly of stars conssts of~ copes of K1. r One of the results ofchen, L and Yeh [1] gves a graceful labelng of B(~, K L t) when e ~< t. We gve here a graceful labelng of B(~K1.,) wth no restrcton on c~ and t. Our graceful labelng gves label 0 to the apex and hence t s possble to combne B(~K1, t) wth SB" to get a graceful labelng of SB"(t, ~) wthout any restrctons on t and ~. Our graceful labelng of B(eK L,) wth no restrcton on ~ and t s gven n table 6. Apex s labeled 0. n fact, after gvng the table 6 we have also ndcated the labelng functon q~ n a compact form.

13 New famles of graceful banana trees 213 l ~ ~o~ -- ~ ~ t f r o 7 o.j.j l ~ ~ o~ ~ o ' ~ txl,, ~9 e~ o = " g~,j ~ o,~ ~'~ ~o~, e-. U~ u~ ~ t o l 0 Z 0 Z

14 214 Vasant N Bhat-Nayak and Ujwala N Deshmukh 96 9Z, " 't /, 63 S & 4.Z /' L & Z, 105 "f ~3 11Z, ~26 ~ Fgure 5 (even). Graceful labelng of 3 - SB 8. he apex gets label L e, Z-, L /, lo& Fgure 5 (odd). Graceful labelng of 3 - SB 7. he apex gets label 7. Refer to table 2 for the graceful labelng of SB" as gven by Chen, L/ and Yeh [1]. We have recast ths graceful labelng n the form of a labelng functon qj after the descrpton of the functon ~b. We now gve an algorthm to get a graceful labelng of SB"(t, 7) where there are no restrctons on t and ~.

15 New.famles of grace[ul banana trees )t21 ~ , / ~ Graceful [abeltlng ot B (5K1,3) Gracefu( [abelt[ng of $ /.t & ' /./. 45 /.6 Fgure 6. Graceful labelng of SB6(3, 5). Algorthm Step 1. dentfy the apex vertces of SB" and B(c~K1. t) to get SB"(t, :0. Gve label n to the apex of SB"(t, :0 thus obtaned. Step 2. Add n to each label of B(~(K1, t) part of SB"(t, :0 except to the apex whch s covered by step 1. Step 3. For all the central vertces of SB" change label x to n - x. Step 4. For all other vertces of SB" change label x to ((n 2 3n)/2) ~(t 1) (n 1) - x. Remark. We do not nclude a table for ths case. he reason s, the edge weghts of the edges of the B(~K1. t) part of SB"(t, :0 reman the same and they are already ndcated n table 6. hough the edge weghts of the SB" part of SB"(t, ~) change, one can refer to table 3 wth f12 = c((t 1) nstead of((n 2 3n)/2) c((t 1). Fgure 6 gves a graceful labelng of SB6(3, 5) obtaned as per Algorthm. he functons c~ and ~O Let v and uj be the lnk vertex and central vertex respectvely of the star K*,, of the banana tree B(~K 1,,), 1 ~<j <~ ~. Let w~ be the end vertex of the star K*'., of ths banana tree for 1 ~<j~< t- 1,1 ~< r ~<cx. he compact form of the labelng functon 4~ s as follows. r forj= 2-1 =t forj = 2 ~b(uj) = ( ~ ) t (~----~) 1 forj=2-1 =q-(t(-1)) forj = 2

16 216 Vasant N Bhat-Nayak and Ujwala N Deshmukh ~p(w~):q--(~)t--(~) to for 1 ~j<<.t- 1 andr=2- l =(-- 1)t ( 1) = t ( - 1). to forl~<j~<t-1 and r=2 Let vj and uj be the lnk vertex and central vertex of the star K1, j nvolved n SB n for 1 ~<j ~< n. Let w~ be the end vertex of the star K1. r of the banana tree SB n, 1 ~<j ~< r - 1, l <~ r <~ n. References (- 1) t~(vj) = (n 1) n ~%)=j (- 1) ~b(w~.) = (n 1) n for j= n -- to fort=n--, l<.j<.n--, l~<.n--. (- l) =(n l)n---- n--( 1) 2 [1] Chen W C, L H and Yeh Y N, Opemfom of nterlaced trees and graceful trees (1990) (R.O.C: ape. awan) preprnt [2] Kotzg A, Recent results and open problems n graceful graphs, Proc. 15th Southeast. Co~.. on Combnatorcs, Graph heory and Computncj (1984) (Baton Rouge, La}: Concjr. Numer [3] Rngel G, Problem 25. heory of graphs and ts applcatons. Proc. Syrup. Smolence (Prague Publ. House of Czechoslovak Academy of Scence) (1964) 162