Pylont Tr Inrns Usn Qurtt Splts By Kvn Ml Htok Blor o Sn Lnor-Ryn Unvrsty 2010 Sumtt n Prtl Fulllmnt o t Rqurmnts or t Dr o Mstr o Sn n Mtmts Coll o Arts n Sns Unvrsty o Sout Croln 2012 Apt y: Év Czrk, Drtor o Tss Anton Sp, Rr Lnoln Lu, Rr Ly For, V Provost n Dn o Grut Stus
Copyrt y Kvn Ml Htok, 2012 All Rts Rsrv.
Dton Ts work s t to my utul né Amn. Your support s lp mk ts tss possl. You nrt mu mor tn your r sr o t strss n prssur tt m lon wt wrtn t. I truly oul not v on ts wtout you. I lov you.
Aknowlmnts Frstly, I woul lk to tnk my vsor Dr. Év Czrk or r un n v. Your lp urn t wrtn o ts tss nnot ovrstt. Ts tss woul not v n possl not or you. I woul lso lk to tnk my ommtt mmrs Dr. Anton Sp n Dr. Lnoln Lu or your toutul sustons, s wll n your tm n ttnton. I xprss my pprton to Dr. Gor MNulty or mkn t pross o wrtn tss mu smplr tn I vr oul v. I woul lso lk to tnk t mmrs o t mt prtmnt t t Unvrsty o Sout Croln. Spl tnks to Knny Brown, Vrn Jonson, Trvs Jonston n Ml Ln or your lp wt ormtn n L A TEX. Your nst sv m mny ours o xtr work. Lst, ut rtnly not lst, I woul lk to xprss my mmns rttu to my mly, splly my motr, Kmrly Htok, my rotr, Kn Htok, n my wonrul né, n soon-to--w, Amn Hur. I oul not v on ts wtout your lov n support. Your nourmnt s rtly rsponsl or my suss. Tnk you ll. v
Astrt Pylont trs r us y olosts n ntsts s wy o lssyn t rltonsps twn rnt txonom unts. T rnn o tr rprsnts on unt volvn nto two or mor rnt unts. T lvs o pylont trs r ll wt t unts tt r to stu. For ny tr, n pylont trs n prtulr, ltn n rsults n prton o t l ll st. W ll su ltons splts n rprsnt t splt tt rsults n prton sts A n B y A B. A splt σ s rlz y tr T tr s n lton o T tt rsults n t prtton xprss y σ. T st o ll splts rlz y T s not Σ(T ). W suss wt t mns or two splts to omptl n v proo o t Splts-Equvln Torm w stts tt or ollton Σ o splts, tr s tr T su tt Σ = Σ(T ) n only t splts o Σ r prws omptl. O ntrst r splts n w ot A n B = 2. Ts r rrr to s qurtts. W sy tt st A o qurtts nrs qurtt s vry tr tt splys A must lso sply s. Su n nrn s ll k-ry nrn wn A = k. A k-ry nrn s ll prmtv, t n not rv rom lowrorr nrns. In s Mstr s Tss Ronstruton Mtos or Drvton Trs, Dkkr rtrz ll prmtv nry n trnry nrns. Stl n Brynt sow tt or ny postv ntr k tr r prmtv k-ry nrns. W prsnt n npnnt rrton o porton o Dkkr s work y lstn ll prmtv nry qurtt splt nrns, s wll s sow tt t only prmtv trnry nrns nvolv < 8 lvs. W lso prov n xmpl o vl trnry nrn. v
Contnts Dton................................. Aknowlmnts............................. v Astrt.................................. v Lst o Furs............................... v Cptr 1 Prlmnrs........................ 1 1.1 Introuton................................ 1 1.2 Funmntl Dntons......................... 4 Cptr 2 Splts............................. 8 Cptr 3 Qurtts n Root Trpls.............. 15 3.1 Qurtts.................................. 15 3.2 Root Trpls.............................. 16 Cptr 4 Bnry Inrns..................... 19 Cptr 5 Trnry Inrns.................... 27 Blorpy................................ 65 v
Lst o Furs Fur 1.1 An X-tr wt X = {1,..., 25}................... 5 Fur 1.2 A root nry pylont tr wt ll st {1,..., 6, ρ}... 5 Fur 1.3 A pylont tr.......................... 6 Fur 1.4 A tr T n t tr T/ tr ontrton......... 7 Fur 3.1 T qurtt........................... 15 Fur 3.2 T root trpl......................... 16 Fur 3.3 Contrtory yl sown tt λ Σ(T )........... 18 Fur 4.1 T tr T lon wt nw vrts s 1, s 2.............. 20 Fur 4.2 Trs T 1 n T 2............................ 22 Fur 4.3 Trs T 1, T 2, n T 3 sr no ommon splts otr tn A..... 22 Fur 4.4 Trs T 1 n T 2 sr no ommon splts otr tn A....... 23 Fur 4.5 Trs T 1, T 2, n T 3 sr no ommon splts otr tn A..... 23 Fur 4.6 Trs T 1, T 2, n T 3 sr no ommon splts otr tn A..... 24 Fur 4.7 Trs T 1, T 2, n T 3 sr no ommon splts otr tn A..... 24 Fur 4.8 Tr sown possl lotons or tonl l......... 24 Fur 4.9 Trs wt l n postons 4, 5, 6, n 7......... 25 Fur 4.10 Tr wt l tt to poston 2............... 26 Fur 5.1 A tr wt qurtt splts A 1,A 2, n A 3.............. 27 Fur 5.2 Ts v trs sr only t splts A 1, A 2, A 3.......... 28 Fur 5.3 T v trs ov v only tr ommon splts........ 29 Fur 5.4 T tr trs ov sr only splts A 1, A 2, n A 3...... 30 v
Fur 5.5 T our trs ov sr only splts A 1 A 2 n A 3........ 30 Fur 5.6 T tr trs ov sr only splts A 1, A 2, n A 3...... 31 Fur 5.7 T v trs ov sr only splts A 1, A 2, n A 3....... 31 Fur 5.8 T our trs ov sr only splts A 1, A 2, n A 3....... 32 Fur 5.9 T our trs ov sr only splts A 1, A 2, n A 3....... 33 Fur 5.10 T v trs ov sr only splts A 1, A 2, n A 3....... 33 Fur 5.11 T our trs ov sr only splts A 1, A 2, n A 3....... 34 Fur 5.12 T two trs ov sr only splts A 1, A 2, A 3 n t ormnton nry nrn...................... 35 Fur 5.13 Ts tr sows tt A 1, A 2, A 3 n yl no trnry nrns.. 36 Fur 5.14 T trs ov sr only splts A 1, A 2, A 3............ 36 Fur 5.15 T trs ov sr only splts A 1, A 2, A 3............ 37 Fur 5.16 T trs ov sr only splts A 1, A 2, A 3............ 38 Fur 5.17 T trs ov sr only splts A 1, A 2, A 3............ 38 Fur 5.18 T trs ov sr only splts A 1, A 2, A 3............ 39 Fur 5.19 T trs ov sr only A 1, A 2, A 3................ 40 Fur 5.20 T trs ov sr only splts A 1, A 2, A 3............ 41 Fur 5.21 T trs ov sr only splts A 1, A 2, A 3............ 41 Fur 5.22 T trs ov sr only splts A 1, A 2, A 3............ 42 Fur 5.23 T trs ov sr only splts A 1, A 2, A 3............ 43 Fur 5.24 T trs ov sr only splts A 1, A 2.A 3............. 44 Fur 5.25 T trs ov sr only splts A 1, A 2, A 3............ 44 Fur 5.26 T trs ov sr only splts A 1, A 2, A 3............ 45 Fur 5.27 T trs ov sr only splts A 1, A 2, A 3............ 45 Fur 5.28 T trs ov sr only splts A 1, A 2, A 3............ 46 Fur 5.29 T trs ov sr only splts A 1, A 2, A 3............ 47 Fur 5.30 T trs ov sr only splts A 1, A 2, A 3............ 47 v
Fur 5.31 T trs ov sr only splts A 1, A 2, A 3............ 48 Fur 5.32 T trs ov sr only splts A 1, A 2, A 3............ 48 Fur 5.33 T trs ov sr only splts A 1, A 2, A 3............ 49 Fur 5.34 T trs ov sr only splts A 1, A 2, A 3............ 50 Fur 5.35 T trs ov sr only splts A 1, A 2, A 3............ 50 Fur 5.36 T trs ov sr only splts A 1, A 2, A 3............ 51 Fur 5.37 T trs ov sr only splts A 1, A 2, A 3............ 51 Fur 5.38 T trs ov sr only splts A 1, A 2, A 3............ 52 Fur 5.39 T trs ov sr only splts A 1, A 2, A 3............ 53 Fur 5.40 T trs ov sr only splts A 1, A 2, A 3............ 53 Fur 5.41 T trs ov sr only splts A 1, A 2, A 3............ 54 Fur 5.42 T trs ov sr only splts A 1, A 2, A 3............ 54 Fur 5.43 T trs ov sr only splts A 1, A 2, A 3............ 55 Fur 5.44 T trs ov sr only splts A 1, A 2, A 3............ 56 Fur 5.45 T trs ov sr only splts A 1, A 2, A 3............ 56 Fur 5.46 T trs ov sr only splts A 1, A 2, A 3............ 57 Fur 5.47 T trs ov sr only splts A 1, A 2, A 3............ 58 Fur 5.48 T trs ov sr only splts A 1, A 2, A 3............ 58 Fur 5.49 T trs ov sr only splts A 1, A 2, A 3............ 59 Fur 5.50 T trs ov sr only splts A 1, A 2, A 3............ 59 Fur 5.51 T trs ov sr only splts A 1, A 2, A 3............ 60 Fur 5.52 T trs ov sr only splts A 1, A 2, A 3............ 60 Fur 5.53 T trs ov sr only splts A 1, A 2, A 3............ 61 Fur 5.54 T trs ov sr only splts A 1, A 2, A 3............ 62 Fur 5.55 T trs ov sr only splts A 1, A 2, A 3............ 62 Fur 5.56 T trs ov sr only splts A 1, A 2, A 3............ 63 Fur 5.57 T trs ov sr only splts A 1, A 2, A 3............ 63 x
Cptr 1 Prlmnrs 1.1 Introuton Bolosts us pylont (volutonry) trs s vsul rprsntton o volutonry vnts n t rltonsps twn rnt txonom unts (sps, nus, mly, t). Wn on txonom unt splts n orms two or mor nw unts, t s rrr to s spton vnt. Bolosts rprsnt spton vnts usn ntrnl vrts o pylont tr. T to rprsnt spton vnts usn trs n wt Crls Drwn s rly s 1837 [6, 7]. Ornlly, t volutonry losnss o rnt sps ws trmn usn pysl omprsons ut t ws sovr tt t pprn o smlr trts not lwys orrlt wt n volutonry nstor s wll s mt xpt. For xmpl, s solly on pprn, t woul sm tt wls r mor losly rlt to srks tn to ppopotmuss ut ts s not t s [3]. It s lso ult to stnus twn sps tt r xtrmly smlr. In som ss, t only wy to urtly stnus twn mmrs o rnt sps s y usn DNA tstn [13]. Wt t snt vnmnts tt m n t son l o t twntt ntury, nlun t us o DNA n t l o noms, olosts wr l to t mu ttr ptur o ow rnt sps r rlt n wr to pl tm on pylont trs [1]. Wt tm, ts lts wll only mprov. A nturl mto or rtn lr pylont tr s to onstrut smllr trs usn only susts o t txonom unts n tn ttmpt to mr ts 1
smllr trs totr [4, 14]. In som ss owvr, tr n svrl rnt wys to mr ts smll trs totr. Wn ts s t s, t s nssry nvor to trmn wn ts sprt rprsnttons r omptl n n omn nto on omprnsv tr. Wl t us o pylont trs s y r t omnnt mto or rprsntn volutonry story, t mtos us to pl t rnt tx on t trs r not wtout ompltons. T morty o olo rltonsps r trmn usn nt mtrl. T mns o trnsrrn nt normton s rrr to s vrtl n trnsr. Vrtl trnsr ours wn nt mtrl s pss rom prnt to osprn. In multllulr ornsms ts s, wt vry w xptons, t only mns o trnsr. Howvr, snl-ll ornsms, n prtulr tr, v mny mtos o wt s rrr to s orzontl n trnsr [11]. Horzontl trnsr ours wn nt mtrl s spr n mnnr tt s not rom prnt to osprn [9]. Btr n sor n xprss orn nt mtrls. Ts s t prmry us o ntot rsstn n tr [5]. T pross o trnsuton nvolvs tr n vruss [15]. Som vruss, ll trops, v t lty to sor trl DNA rom ost ornsm n trnsr tt DNA to notr trum, tt my not los rltv o t ost [5]. Som tr lso v t lty to trnsr nt mtrl rtly y mns o ll-to-ll ontt. In ts wys, ssntlly unrlt ornsms n sply t sm ns. Amon mor omplx ornsms, tr r lso svrl ults tt rsult rom usn nt mtrls to nr volutonry rltonsps. Hyrzton n our trouout svrl rnt txonom lvls. Intrrn twn rnt susps wtn t sm sps s rly ommon n ntur: n xmpl n mtn twn Bnl tr n Srn tr [18]. Hyrs n orm twn rnt sps wtn t sm nus. Muls r t rsult o mtn twn onky n ors. Intrsps yrzton s lso n oun mon r- 2
nt sps o yks n squrrls [16]. Altou rr, yrs my lso orm twn mmrs o two rnt nr wtn t sm mly. Wl n nrl, t rtr prt t two ornsms r ntlly, t lowr t ns o sussul yr, su ntrnr yrs o our. T Lyln yprss s rsult o t rossn o Montry yprss wt n Alsk yprss [19]. In mny ss, splly wt ornsms tt r not losly rlt, ts yrs tn to unl to rprou, ut ts s not lwys t s [7]. Convrnt voluton s notr prolm wn onsrn t voluton o omplx ornsms. Plnts, nmls, un v so mny trts, t s not out o t rlm o posslty tt som smlr trts my volv npnntly n rtr unrlt tx. Wn lns tt r not losly rlt vlop smlr trts npnntly, t s ll onvrnt voluton [12]. A prm xmpl nvolvs rs n ts. T most rnt ommon nstor o rs n ts not v wns, yt ot v vlop tm ovr tm. Convrnt voluton n lso our t t DNA lvl wt rnt ornsms vlopn smlr nzyms n protns tt sply s smlr trts npnntly o otr [10]. Wl nt mtrls n normton r rly vll or lvn ornsms, olosts must tr normton out xtnt sps usn t ossl ror. Ts lrly rsults n lr mount o lost normton. It s vnt tt som ornsms tt v om xtnt, tr wr not osslz or v yt to sovr. Evn mon t ornsms tt v n sovr, only r tssus r unntly osslz; ltou vnmnts r llown or t nvrstton o som sot tssus [2]. T rtr k n tm n ornsm lv, t lss normton n n rom t rmns. Lonr-xtnt sps tn to v mor muous volutonry stors. Evn wt ts ults, t l o pylonts s t lty to strtn t tnl volutonry story o t ornsms on Ert. In ts tss, w ntn 3
to monstrt ow t o pylont trs n us to lp vsulz t rltonsps tt olosts sovr trou t us o DNA squnn n noms. W n y nn som unmntl s n rp tory. W tn xplor t o splts n tr n wys n w ts splts n us to ronstrut t tr. Two mportnt typs o splts (qurtts n root trpls) r xpoun upon sprtly n Cptr 3. Cptrs 4 n 5 monstrt ow on n us n nomplt st o qurtt splts to nr otr omptl qurtts. 1.2 Funmntl Dntons As wt ny sut, n orr to unrstn t omplx, on must v rsp o t s. W v svrl ssntl ntons n ts ston. For rrs wo sr mor n pt ntrouton, s Introuton to Grp Tory y Truu [17]. Dnton 1.1. A tr s onnt rp tt ontns no yls. Dnton 1.2. A tr n w som vrts (nlun vry vrtx o r on or two) r ll wt sont susts o st X s ll n X-tr. Not tt vrts o r rtr tn two my or my not ll. Exmpl 1.3. Fur 1.1 low splys n X-tr wr X = {1,..., 25}. Not tt tr r som vrts o r rtr tn 2 tt r ll n tt som vrts r ll y snl lmnts o X. Dnton 1.4. A tr tt s no vrts o r two n s l unquly lll y snlton sust o t st X s ll pylont tr or pylont X-tr. Dnton 1.5. A root pylont tr s pylont tr tt s n ntrnl vrtx tt s stnus n s ll t root. T root s not y ρ n my v r two. All ntrnl vrts otr tn t root must v r rtr 4
3, 6, 11 5, 21 10, 13, 19 20, 22 7 1, 2, 4 16, 17, 18 14 8, 15 9 24, 25 12, 23 Fur 1.1: An X-tr wt X = {1,..., 25} tn or qul to tr. In ts tss t trm pylont tr wll rr to t unroot s. Dnton 1.6. A tr n w vry ntrnl vrtx s r tr s ll nry tr. A nry pylont tr n root nry pylont tr, sown n Fur 1.2, r n smlrly. In root nry pylont tr owvr, t root stll s r two. ρ 1 3 4 5 6 2 Fur 1.2: A root nry pylont tr wt ll st {1,..., 6, ρ} Dnton 1.7. Lt t ll st o n X-tr T not L(T ). T lton o ny o T rsults n xtly two smllr sutrs T 1 n T 2 o T. Ts lton lso prttons L(T ) nto two susts L(T 1 ) n L(T 2 ) wr L(T 1 ), L(T 2 ) n L(T 1 ) L(T 2 ) = L(T ). W ll ts prtton o L(T ) splt o T. Splts r 5
somtms ll X-splts. W wll not y Σ(T ), t ollton o ll t splts o T orm n ts son. W ll Σ(T ) t splt st o T. T splt tt rsults n t prtton o L(T ) nto sts A n B wll not A B. Not tt t splt A B s t sm s t splt B A. Exmpl 1.8. Consr t tr low n Fur 1.3. T splt orrsponn to ltn 1 s {1, 2} {3, 4, 5, 6} wl t splt orrsponn to 2 s {1, 2, 3, 4} {5, 6}. 5 6 1 2 2 1 3 4 Fur 1.3: A pylont tr Dnton 1.9. Lt T tr n n o T wt n vrts u n v. T nw tr T = T/ s t tr orm y t ontrton o. T tr s orm y rpln vrts u n v wt t snl vrtx ν su tt N(ν) = (N(u) \ {v}) (N(v) \ {u}). Exmpl 1.10. T trs n Fur 1.4 monstrt n ontrton. T tr on t rt s t ontrt. It s mportnt to not tt ontrtn n, rmovs splt rom t tr. Wn onsrn pylont trs, ontrtn n rsults n loss o rsoluton. It tks two sprt spton vnts n ons tm nto on. It s tn mpossl to stnus w vnt ppn rst. For olosts, t st s snro or rprsntn volutonry t woul root nry pylont tr. Ts tr woul v root tt orrspons to t ommon nstor rom w ll o t lvs volv. E ntrnl vrtx o t 6
1 2 8 4 6 10 2 8 1 4 6 10 9 3 11 5 7 9 3 11 5 7 Fur 1.4: A tr T n t tr T/ tr ontrton tr woul v r tr n woul orrspon to t rton o xtly two nw txnom unts. A tr tt splys ll t normton possl or ts st o l lls s s to ully rsolv. Sn normlly t normton ollt out t volutonry rltonsps mon vn st o tx s not prt, t nnot xpt tt ll t sts rsult n ully rsolv trs. Wn tr s not ully rsolv, tr r vrts o r 4 or r. Ts orrspons to spton vnt tt rsult n 3 or mor nw tx. Sn ts s rrly t s n ntur, t s ly lkly tt ts vrtx o r 4 rprsnts two or mor sprt spton vnts wos orr nnot trmn. 7
Cptr 2 Splts Ts ptr wll suss splts on t ll sts o pylont trs. Dntons wll nlu: splt, splt st, omptlty o splts, omptlty o trs, ontrton n nu sutr. Torms wll nlu: Splts-Equvln Torm. Dnton 2.1. Lt T n unroot pylont tr n A sust o L(T ). Lt T (A) t mnml sutr o T tt nlus lmnt o A. Supprssn ll vrts o T (A) wt r two rsults n t sutr o T nu y A w s not T A. I T s root pylont tr, tn stnus t vrtx o T (A) tt ws losst to t root n supprss ny otr vrts o r two n orr t orm T A. Inu sutrs r ll rstrt trs n Smpl-Stl. Dnton 2.2. Lt T n S pylont trs. Tn w sy tt T s omptl wt S S n orm y ontrtons o n nu sutr o T or S s n nu sutr o ontrton o T. W not tt T s omptl wt S y S T. Proposton 2.3. T rlton s prtl orr Proo. Lt T tr. Clrly T T sn t nu sutr T L(T ) = T lon wt t mpty st o ontrtons n rsults n T. Now suppos tt S T n T S. Tn L(S) L(T ) n L(T ) L(S), n L(S) = L(T ). Also, w know tt S s T wt ll r two vrts supprss. But, sn T ws pylont tr, t no r two vrts rom t nnn. So w s tt S = T. Now suppos tt or pylont trs S, T, U w know tt S T n T U. 8
Tn L(S) L(T ) L(U). Now, T s orm rom ontrtons o s rom U n S s orm rom ontrtons o T. Hn S s orm rom ontrtons o U n w s tt S U. So w v sown tt s n t prtl orr. T prnpl rsult o ts ston wll proo o t Splts-Equvln Torm rst prov y Bunmn n 1971. W wll rst suss svrl ntons n lmmt to n t proo o t torm. Dnton 2.4. T pr o splts A 1 B 1 n A 2 B 2 r s to omptl ny o t ollown ntrstons s mpty A 1 A 2, A 1 B 2, A 2 B 1, B 1 B 2. I st Σ o splts s su tt pr o splts n Σ r omptl, w sy tt Σ s onsstnt. Dnton 2.5. A splt o t orm A B wr mn{ A, B } = 1 s ll trvl splt. For nt st X, trvl splt s o t orm {x} {X \ {x}} wr x s n lmnt o X. Proposton 2.6. A trvl splt o st X s omptl wt vry X-splt. Proo. Lt x X. Tn {x} {X \ {x}} s t trvl splt orrsponn to x. Lt A B notr X-splt. Not tt tr x A or x B ut not ot. Wtout loss o nrlty, sy x A. Tn or t two splts {x} {X \ {x}} n A B w s tt {x} B = n rom Dnton 2.4 w know tt ts two splts r omptl. Lmm 2.7. T s pylont X tr n only T splys t st Σ trv (X) o trvl splts o X. 9
Proo. ( ) Lt T pylont X tr or som nt st X. From t nton o pylont tr (Dnton 1.4) w know tt vry l s ll unquly y snl lmnt o st X. T splts orrsponn to ltn t pnnt s o T r prsly t trvl splts Σ trv (X). ( ) Lt T n X tr tt splys Σ trv (X). Tn or x X tr s n su tt T rsults n t splt x {X \ {x }}. Sn x s splt rom t rst o X t s lr tt som vrtx o T s ll y t snlton st {x }. W n to sow tt ts vrtx s l. Suppos not. Tn ts s n ntrnl vrtx n t orrspons to rnn o T. Ts must tn l to on or mor lvs o T. But sn T s n X tr, ll vrts o r on (lvs) must ll. Suppos L s t st o lls rom ts lvs. Tn T woul rsult n t splt {{x } L} {X \ {{{x } L}} w s ontrton. So w s tt tr s l tt s ll y t snlton st {x } or vry x X n tror T s pylont X tr. Lmm 2.8. (Ronson n Fouls 1981) Lt T n X-tr n lt σ 1 n σ 2 lmnts o Σ(T ) su tt σ 1 σ 2. Tn X n prtton nto tr sts X 1, X 2, X 3 su tt σ 1 = X 1 (X 2 X 3 ) n σ 2 = (X 1 X 2 ) X 3. Furtrmor, X 1 X 3 =. Proo. Lt T n X-tr n σ 1, σ 2 Σ(T ). By nton, σ 1 orrspons to T \ 1 wr 1 = {u 1, v 1 } s n o T onntn vrts u 1 n v 1. Smlrly, σ 2 orrspons to T \ 2 wt 2 = {u 2, v 2 }. Sn σ 1 σ 2 w s tt 1 n 2 r lso stnt. Sn T s tr, tr xsts unqu pt rom u 1 to u 2 n T. Not tt u 1 u 2 ut t my t s tt v 1 = v 2. Etr wy, w n s tt X n v nto tr susts, X 1, X 2, X 3 s ollows. Consr t omponnts o T \ { 1, 2 }. Lt C 1 t omponnt tt nlus u 1, C 2 t omponnt tt 10
nlus v 1 n C 3 t omponnt tt nlus u 2. Sttn L(C ) = X, w v oun t sr sts. Exmpl 2.9. Consr t tr rom Fur 1.3, tt s rprou low. Lt σ 1 t t splt T \ 1 n σ 2 T \ 2. Dnn X 1 = {1, 2}, X 2 = {3, 4}, n X 3 = {5, 6} rsults n t prtton sr ov. W s tt σ 1 = X 1 (X 2 X 3 ) n σ 2 = (X 1 X 2 ) X 3. Also, X 1 X 3 =. 5 6 1 2 2 1 3 4 Now onsr ny tr T. Lt unton rom nt st X nto t vrtx st V (T ). In otr wors, vrtx o T s ll y on or mor lmnts rom X n tus T s n X-tr. Color t lmnts o X tr r or lu. W now olor t vrts o t tr s on ts olorn o X n t ollown wy. Lt v V (T ) n lmnt o (X). I vry lmnt o 1 (v) s t sm olor, tn ssn ts olor to v. I tr r lmnts n 1 (v) o ot olors, tn olor v r n lu. Ts olorn o t vrtx st s rrr to s t olorn o V nu y. W sy tt surp T o T s monoromt ll o t vrts n V (T ) v t sm olor. Lmm 2.10. Lt T tr n unton rom nt st X nto t vrts o T. Consr t olorn o V (T ) nu y s sr ov. Now, suppos tt or E(T ) tt prsly on omponnt o T \ s monoromt n t nu olorn. Tn tr xsts unqu vrtx v V su tt vry omponnt o T \ v s monoromt. 11
Proo. Frst w prov t xstn o su vrtx. Bn y ssnn n orntton to o T wy rom t monoromt omponnt o T \. Cll ts ornt rp T. Wt ts orntton, tr must vrtx v V (T ) su tt t outr o v s 0. I ts wr not t s tn vry vrtx woul v t lst on rt wy rom t n w oul onstrut n nnt rt pt n T. Now, w onsr T \ v w s tt vry omponnt s monoromt. Tus w v oun vrtx tt stss t proposton. Now w wll sow tt ts vrtx n unqu. Suppos ts s not t s. Tn tr r two stnt vrts v, v V (T ) su tt v n v ot v t sr proprty. Sn T s tr, tr s unqu pt P tt onnts v to v. Sn v v ts pt P s t lst on lon. Slt ny o P. Tn rom t sttmnt o t proposton, xtly on omponnt o T \ s not monoromt. Assum wtout loss o nrlty tt ts s t omponnt ontnn vrtx v. Ts ls to ontrton. W ssum tt vry omponnt o T \ v ws monoromt ut w v ust sown tt t omponnt ontnn vrtx v s not. So w v sown tt su vrtx xsts n s unqu. Lmm 2.11. Lt A B n X-splt n lt T n X-tr so tt A B s not splt o T ut A B s omptl wt wt vry splt n Σ(T ). Tn tr s unqu vrtx v o T su tt, or omponnt T o T \ v tr L(T ) A or L(T ) B. Proo. Lt : X V (T ) t unton tt ssns t lls to t vrts o T. Color t lmnts o A r n t lmnts o B lu n onsr t olorn o V (T ) nu y. Now suppos tt A 1 B 1 s splt o T. Tn sn A 1 B 1 s omptl wt A B w know tt on o t ollown ntrstons s mpty A A 1, A B 1, B B 1, B A 1. 12
Wtout loss o nrlty, suppos tt B B 1 =. Tn, sn A n B prtton V (T ) w n s tt B 1 A n n B 1 s r. Sn t splt A 1 B 1 orrspons to ltn som 1 o T, w know tt T \ 1 s monoromt omponnt. I A 1 wr sust o B n B 1 wr sust o A, tn t woul t s tt A B = A 1 B 1 w woul ontrt t ssumpton tt A B / Σ(T ). Now w s tt T \ 1 s xtly on monoromt omponnt or E(T ) n w n ppl to Lmm 2.10. So w know tt tr xsts unqu vrtx v V (T ) su tt vry omponnt o T \ v s monoromt. In otr wors, omponnt T o T \ v s tr ompltly r or ompltly lu n n L(T ) A or L(T ) B. W wll now prov t Splts-Equvln Torm s stt low. Torm 2.12. (Splts-Equvln Torm) Lt Σ ollton o X-splts. Tn, tr s n X-tr T su tt Σ = Σ(T ) n only t splts o Σ r prws omptl. Furtrmor, su n X-tr T xsts, tn up to somporpsm, T s unqu. Proo. Suppos tt Σ s ollton o X-splts nu y t s o som X-tr. Lt σ 1 n σ 2 stnt lmnts o t splt st Σ. W know rom Lmm 2.8 tt tr must prtton o X nto sts X 1, X 2, X 3 su tt σ 1 = X 1 (X 2 X 3 ) n σ 2 = (X 1 X 2 ) X 3. Sn X 1 X 3 =, w n s rom Dnton 2.4 tt t splts σ 1 n σ 2 r omptl. Tror ll t splts o Σ must prws omptl. Now, suppos tt Σ s ollton o prws omptl X-splts. W wll sow y nuton on t rnlty o Σ to sow tt Σ = Σ(T ) or som tr T n lso tt, up to somorpsm, ts tr T s unqu. Bs Cs: Suppos tt Σ = 0. Tn, up to somorpsm, tr s unqu X-tr T or w Σ = Σ(T ). In ts s, w s tt T t tr onsstn o on vrtx ll y ll o X. Inuton Stp: Now suppos tt Σ = k + 1 wr k 0 n tt t torm s tru or 13
Σ = k. Now lt A B n lmnt o Σ. Sn Σ s st o prws omptl X-splts w s tt Σ A B must lso. So y t nuton ypotss, tr xsts, up to somorpsm, unqu tr T wr Σ A B = Σ(T ). Lt t unton : X V (T ) ll t vrts o T. Usn Lmm 2.11, w s tt tr s vrtx v V (T ) su tt or vry omponnt T o T \ v, tr L(T ) A or L(T ) B. W now rt tr T n t ollown mnnr. Rpl v n T y two nw vrts v A n v B so tt {v A, v B } E(T ). Now tt t vrts o (A) to t nw vrtx v A n tt t vrts o (B) to v B. Now w n nw unton s ollows (x) = (x) (x) v v A v B (x) = v n x A (x) = v n x B Ts nw unton lls t vrts o our nw tr T. It s lr tt T s n X- tr. Any splt o T s lso splt o T n lso, y onstruton T \ {v A, v B } = A B. So w v onstrut tr T so tt Σ = Σ(T ). By t nuton ypotss, w know tt T s t only tr, up to somporpsm, tt splys Σ A B. It ollows tn, tt, up to somorpsm, T s t only tr tt s splt st Σ. 14
Cptr 3 Qurtts n Root Trpls 3.1 Qurtts Ts ptr wll suss qurtt splts n root trpls. Dntons wll nlu: qurtt, root trpl, spn, qurtt st n root trpl st. Proos o svrl lmms on tr omptlty wll vn. Dnton 3.1. An unroot nry tr wt our lvs s ll qurtt. W wll not t qurtt wt l prs {, } n {, } on y snl ntrnl s. W us ts notton us ltn t snl non-pnnt rsults n t splttn o t two vrtx prs. For ts rson, qurtts r somtms rrr to s qurtt splts. Fur 3.1: T qurtt Dnton 3.2. For st Q o qurtts, t spn o Q, w w not Q, s t st o ll unroot trs tt r omptl wt qurtt n Q n v lvs lll y L(Q). 15
Dnton 3.3. For n unroot tr T t qurtt st q(t ) s t st o ll qurtts tt r nu sutrs o T. Tt s, t st o qurtts tt n orm y ontrtn s o nu sutrs o T. 3.2 Root Trpls Ts ston wll suss root trpls Dnton 3.4. A root nry tr wt xtly tr lvs s ll root trpl. W wll not t root trpl wt l st {, } onnt y t root t l y. ρ Fur 3.2: T root trpl Dnton 3.5. For st R o root trpls, t spn o R, w w not R s t st o ll root trs tt r omptl wt root trpl n R n v lvs ll y L(R). Dnton 3.6. For root tr T t root trpl st r(t ) s t st o ll root trpls tt r nu sutrs o T. Tt s, t st o root trpls tt n orm y ontrtn s o nu sutrs o T. Rll rom nton 2.2 tt or two trs S n T, w sy S T S s orm y ontrtons o n nu sutr o T or S s n nu sutr o ontrtons o T. W sow n proposton 2.3 tt s prtl orr. Blow, n Torm 3.2 w v wy to trmn, or trs S n T, T s omptl wt S,.. S T. 16
Lmm 3.7. For pylont tr S, λ λ s splt o S, wt mn{ λ, λ } 2 tn s qurtt splt o S or ll prs, λ n, λ. Proo. Sn λ λ s splt o S, tr s n S so tt S prous t splt λ λ wr λ λ = n λ λ = L(S). Lt, λ n, λ, tn lrly, S rsults n λ n λ. Tror S rsults n t qurtt splt n w s tt q(s). Lmm 3.8. I T s pylont tr n q(t ) or ll prs {, } λ n {, } λ, tn λ λ s splt o T. Proo. Lt T pylont tr. Lt, lmnts o λ n lt u 0 t vrtx t w t tr splts nto t sprt rns ln to n. Hr w us nuton on λ. Bs Cs: I λ = 2 tn lrly λ λ Σ(T ). Inuton Stp: Assum tt λ = n n t sttmnt s tru or ll sts o sz < n. Lt x 0 λ. Tn, y t nuton ypotss, t splt { λ \ {x 0 }} s n Σ(T ) sn ts st s o sz on lss tn λ. Ts mns tt tr s n 0 = {u 0, v 0 } so tt T \ 0 prous t ormnton splt. Also, or x λ tr s n wos rmovl rsults n x 0 x. Consr on su 1. Suppos tt 0 1. Tn 1 = {u 0, v 1 }. But now, u 0 v 1 x 1 v 0 u 0 s yl ns T (sown n Fur 3.3) w s ontrton sn T s tr. So w s tt or ll, t must tt = 0 n tt rmovn 0 rom T rsults n λ. Tr must su n or pr o vrts n λ n smlr rumnt sows tt λ λ must splt o T. Lmm 3.9. For pylont trs S n T on t sm ll st L, Σ(S) Σ(T ), tn S T. Proo. Lt T n S pylont trs on t sm ll st L su tt Σ(S) Σ(T ). For E(S), S rsults n som splt α β. Sn Σ(S) Σ(T ), 17
t splt α β must orrspon to T or som E(T ). Howvr, tr my splts n Σ(T ) tt v no orrsponn n E(S). So w n s tt S s orm y ontrtn xtly ts s o T. So S s orm y ontrtosn on t s o t nu tr T L n tror S T. v 1 x 0 u 0 0 v 0 1 x 1 λ \ {x} Fur 3.3: Contrtory yl sown tt λ Σ(T ) Torm 3.10. Lt S n T unroot pylont trs. T s omptl wt S (.. S T ) n only q(s) q(t ) n L(S) L(T ). Proo. ( ) Assum tt S T. I t splt s n q(s), tn surly S s omptl wt t qurtt wt l sts {, } n {, } sn t qurtt n orm y ontrtn s o S {,,,}. W sow n Proposton 2.3 tt s trnstv n so t qurtt S n S T mpls tt T. Tror q(t ). Sn S T y ssumpton, t s lr tt L(S) L(T ). ( ). Suppos tt q(s) q(t ) n L(S) L(T ). Clrly, T L(S) s omptl wt S tn T s lso omptl wt S. So w n rstrt ourslvs t s wn L(S) = L(T ). Lt λ λ splt o S. By Lmm 3.7, w s tt q(s) or ll, λ n, λ. From t ssumpton, w s tt ts mns q(t ) or ll su prs n tror λ λ Σ(T ) y Lmm 3.8. Ts ounts or vry splt λ λ wr mn{ λ, λ } 2. I, wtout loss o nrlty, λ = 1, tn t splt λ λ s trvl splts. W know rom Lmm 2.7 tt sn S n T r ot pylont trs, ty ot sply ll t trvl splts. So w s tt Σ(S) Σ(T ). Sn w rstrt T to L(S) w n ppl to Lmm 3.9 to sow tt S T. 18
Cptr 4 Bnry Inrns Dnton 4.1. W sy tt st o qurtt splts Q nrs t qurtt splt s wn vry tr T tt splys vry splt n Q lso splys s. I Q nrs s t s not Q s. Wn Q = k ts r ll k-ry nrns. I Q s or som Q o sz k n tr s no propr sust o Q tt nrs s, ts s ll prmtv k-ry nrn. Brynt n Stl sow tt tr r prmtv k-ry nrns or ll k [4]. In s Mstr s Tss rom 1986, Ronstruton Mtos or Drvton Trs, M.C.H. Dkkr numrt ll nry n trnry qurtt splt nrns[8]. Hs rsults wr not vll to m n I v tror not sn ny o Dkkr s work. Wt ollows s n npnnt r-rton o porton o s Mstr s Tss. Lt A st o two qurtt splts A 1 n A 2. Lt s qurtt splt su tt s / A. W ws to know unr wt rumstns os A s. W lt t ll st L(A) o A {L(A 1 ) L(A 2 )}. Lmm 4.2. I L(A) L(s) < 2 tn A s. Proo. Cs 1. Lt L(A) L(s) = 0 n lt T tr tt splys ot splts n A. Lt s t splt wr {,,, } L(A) =. Now oos som ntrnl vrtx v o T n tt our nw s to v wt on ln to o t our nw vrts {,,, }. Cll ts nw tr T. Sn,, n r ll nt to t vrtx v tr s no wy to splt, rom, n T. It s lr tt q(t ) q(t ) n 19
tror T splys A ut T os not sply s. Sn w v rt tr tt splys A n not s w s tt A s. Cs 2. Lt L(A) L(s) = 1 n lt T tr tt splys ot splts n A. Lt s t splt wr {,,, } L(A) = {} wtout loss o nrlty. Coos n ntrnl vrtx v o T n tt tr nw s to v wt on ln to o t nw vrts {,, }. An, tr s no splt tt sprts {, } rom {, }. T s n xmpl o tr tt splys A ut not s. So w s tt A s. Lmm 4.3. I L(A) L(s) = 2 tn A s. Proo. Lt L(A) L(s) = {, } wtout loss o nrlty. So w v tt L(s) = {,, s 1, s 2 }. Lt T tr tt splys ot splts n A. I w s 1 n s 2 to T n su wy tt s 1 n s 2 r nt to t sm ntrnl vrtx o T, tn tr n no splts o t orm v 1 s 1 v 2 s 2 wr v 1, v 2 L(A) s sn n Fur 4.1. s 1 s 2 Fur 4.1: T tr T lon wt nw vrts s 1, s 2 So suppos tt s s o t orm v 1 v 2 s 1 s 2. A s 1 to T so tt t s nt to v 1 n s 2 so tt t s nt to v 2. W s tt T {s 1, s 2 } nnot sply t splt s. Tr r no mor posslts or t rrnmnt o s n so w s tt wn L(A) L(s) = 2, A q. 20
Lmm 4.4. I L(A) L(s) = 3 tn A s. Proo. Suppos tt L(A) L(s) = {,, }. Tn wtout loss o nrlty, lt s t splt s 1. Lt T tr tt splys ot splts n A. Sn s 1 / L(A) w s tt s 1 n so tt t s nt to ny ntrnl vrtx o T. In prtulr, s 1 n so tt t s nt to vrtx. Sn s nt to s 1 tr s no splt tt wll sprt rom s 1. Hn or ny splt s w s tt A s. Not tt w v now sown tt A s tn t must t s tt L(A) L(s) 4. Sn L(s) = 4, w s tt wn A s t must tt L(q) L(A). In orr or n nrn to m, w s rom nton 4.1 tt ll trs splyn A must lso sply s. Tt s, T s t st o ll trs T tt sply A, n A s, tn s q(t). It s lr tn tt som numr o trs n T r sown to v no ommon qurtt splts otr tn A tn t s n sown tt tr r no nrns to m rom A. Torm 4.5. T ollown s n xustv lst o nry nrns nvolvn qurtt splts. T st {,,,, } rprsnts t lvs o tr T tt splys t qurtt splts.,,,, 21
Proo. W n s tt L(A) {5, 6, 7, 8}. W wll prov t torm y ss. Cs 1. Suppos tt L(A) = 8.. L(A 1 ) L(A 2 ) = 0. Wtout loss o nrlty lt A 1 n A 2. Fur 4.2: Trs T 1 n T 2 W n s rom Fur 4.2 tt t trs T 1 n T 2 ot sply t splts o A ut v no otr splts n ommon. From ts w n s tt t splts o A o not nr ny otr qurtt splts. Cs 2. Suppos tt L(A) = 7.. L(A 1 ) L(A 2 ) = 1. Wtout loss o nrlty lt A 1 n A 2. Fur 4.3: Trs T 1, T 2, n T 3 sr no ommon splts otr tn A. In Fur 4.3, w v tr trs tt ll sply ot A 1 n A 2. I t wr t s tt A nr notr qurtt splt s, tn T 1, T 2, n T 3 woul ll sply s. Sn tr r not splts tt r ommon to ll tr trs, otr tn tos n A, w n s tn no nrns n m wn L(A) = 7. 22
Cs 3. Suppos tt L(A) = 6.. L(A 1 ) L(A 2 ) = 2. Hr w must look t svrl suss or tr r our possl rrnmnts o t lvs n w L(A) = 6. Sus 3.1. Suppos tt A 1 s t qurtt splt n A 2 s t qurtt splt. It s lr rom Fur 4.4 tt ts two splts o not nr tr. Fur 4.4: Trs T 1 n T 2 sr no ommon splts otr tn A. Sus 3.2. Sus 3.2 Suppos tt A 1 s t qurtt splt n A 2 s t qurtt splt. As sn n Fur 4.5, tr r no splts tt n nr rom A 1 n A 2. Fur 4.5: Trs T 1, T 2, n T 3 sr no ommon splts otr tn A. Sus 3.3. Suppos tt A 1 s t qurtt splt n A 2 s t qurtt splt. As vn y t trs n Fur 4.6, t splts n A o not nr tr qurtt splt. Sus 3.4. Suppos tt A 1 s t qurtt splt n A 2 s t qurtt splt. It s sown n Fur 4.7 tt no nrns n m rom ts two qurtt splts. 23
Fur 4.6: Trs T 1, T 2, n T 3 sr no ommon splts otr tn A. Fur 4.7: Trs T 1, T 2, n T 3 sr no ommon splts otr tn A. W v now xust ll posslts or two qurtt splts A 1 n A 2, or w L(A 1 ) L(A 2 ) 6. T only rmnn s nvolvs splts twn w only on l n rnt. Cs 4. Now, ssum tt L(A 1 ) L(A 2 ) = 3.. L(A) = 5. Wtout loss o nrlty, lt A 1 t splt. In ts snro, w n onsr 4-l tr T tt splys qurtt splt A 1 n xplor t possl lotons or our t l. Consr Fur 4.8 low. 1 6 2 3 4 5 7 Fur 4.8: Tr sown possl lotons or tonl l 24
Tr r 7 possl lotons to w t t l n to T. W wll onsr son qurtt splt A 2, wt L(A 2 ) = {,,, }, n trmn w o ts postons woul vl lotons or t nw l. W wll tn trmn tr r nrns to m rom A 1 n A 2. Sus 4.1. Suppos tt A 1 s t qurtt splt s ov, n A 2 s t qurtt splt. To orm nw tr T = T + {} tt wll sply A 1 s wll s A 2, w s tt n tt t postons 4, 5, 6, or 7. I wr to postons 1, 2 or 3, t rsultn tr T woul not sply A 2. Now lt us onsr t trs n Fur 4.9. By onstruton, o ts trs splys splts A 1 n A 2 n w trmn usn Fur 4.8 tt ts r t only trs tt o so. Not, owvr, tt o ts trs lso splys t qurtt splt. Fur 4.9: Trs wt l n postons 4, 5, 6, n 7 Sn vry tr tt slys A lso splys, w v sown tt,. Ts s t rst o t our nry nrns. 25
Sus 4.2. An, w lt A 1 t splt. I A 2 s t splt tn, rom Fur 4.8, w s tt t only possl loton or l s n poston 2. Fur 4.10: Tr wt l tt to poston 2 In Fur 4.10 w s tt t only tr tt splys A 1 n A 2 lso splys tr otr splts. So rom ts tr w n trmn t ollown nry nrns,,, As tr r no otr posslts or A 1 n A 2 w v sown tt ts our nry nrns r prsly t nrns tt n rwn rom two qurtt splts. 26
Cptr 5 Trnry Inrns W ontnu t work o t prvous ptr, ts tm lttn A = {A 1, A 2, A 3 }. Sn t sz o A s now 3, w rr to ny nrns oun rom su st trnry nrns. Lmm 5.1. I L(A) = 12 tn tr s no qurtt splt s su tt A s. Proo. Wtout loss o nrlty, lt A 1 =, A 2 =, n A 3 = kl. T proo o t ov lmm s lr rom Fur 5.1 low. W n s tt sn t prs {} n {} n ntrn n t prs {} n {kl} n ntrn tt tr r no nrns tt n rwn rom t tr low. k l Fur 5.1: A tr wt qurtt splts A 1,A 2, n A 3 Lmm 5.2. I L(A) = 11 tn tr r no nrns tt n m. Proo. Wtout loss o nrlty w n lt A 1 =, A 2 =, n A 3 = k. I w n n ollton o trs tt sply o ts tr splts ut v 27
no otrs n ommon, tn w v prov t lmm. Consr Fur 5.2 low. Ts v trs sply A 1, A 2, A 3 ut v no otr splts ommon to ll v. k k k k k k Fur 5.2: Ts v trs sr only t splts A 1, A 2, A 3 Lmm 5.3. I L(A) = 10 tn tr r no nrns tt n m. Proo. Wn L(A) = 10 w r or to onsr two ss, o w l to svrl suss. W n s tt wt ll st o sz 10, tt tr r two rnt wys to oos our rst two splts. Wtout loss o nrlty, lt A 1 =. Now, A 2 must su tt L(A 1 ) L(A 2 ) {0, 1, 2} n orr to t 10 stntly ll lvs. Not owvr, tt L(A 1 ) L(A 2 ) = 2, tn sn tr r 10 totl lvs, t must tt L(A 1 ) L(A 3 ) = 0. Sn w n rtrtly snt w splt s A 2 n w s A 3 ts s ssntlly t sm s t s wn 28
L(A 1 ) L(A 2 ) = 0. Ts rlzton llows us to only onsr t two ss wn L(A 1 ) L(A 2 ) {0, 1}. Cs 5. Suppos tt L(A 1 ) L(A 2 ) = 0. Tn, wtout loss o nrlty, A 1 = n A 2 =. Hr w onsr possl rrnmnt o A 3. Sus 5.1. W v t ollown tr splts: A 1 = A 2 = A 3 = W n s rom Fur 5.3 low tt ts tr splts o not nr ourt. Fur 5.3: T v trs ov v only tr ommon splts Sus 5.2. W v t ollown tr splts: A 1 = A 2 = A 3 = W n s rom Fur 5.4 low tt ts tr splts o not nr ourt. 29
Fur 5.4: T tr trs ov sr only splts A 1, A 2, n A 3 Sus 5.3. W v t ollown tr splts: A 1 = A 2 = A 3 = W n s rom Fur 5.5 low tt ts tr splts o not nr ourt. Fur 5.5: T our trs ov sr only splts A 1 A 2 n A 3. 30
Sus 5.4. W v t ollown tr splts: A 1 = A 2 = A 3 = W n s rom Fur 5.6 low tt ts tr splts o not nr ourt. Fur 5.6: T tr trs ov sr only splts A 1, A 2, n A 3 Sus 5.5. W v t ollown tr splts: A 1 = A 2 = A 3 = W n s rom Fur 5.7 low tt ts tr splts o not nr ourt. Fur 5.7: T v trs ov sr only splts A 1, A 2, n A 3 31
Sus 5.6. W v t ollown tr splts: A 1 = A 2 = A 3 = W n s rom Fur 5.8 low tt ts tr splts o not nr ourt. Fur 5.8: T our trs ov sr only splts A 1, A 2, n A 3 Cs 6. Now suppos tt L(A 1 ) L(A 2 ) = 1. Tn, wtout loss o nrlty, A 1 = n A 2 =. Hr w onsr possl rrnmnt o A 3. Sus 6.1. W v t ollown tr splts: A 1 = A 2 = A 3 = W n s rom Fur 5.9 low tt ts tr splts o not nr ourt. Sus 6.2. W v t ollown tr splts: A 1 = A 2 = A 3 = W n s rom Fur 5.10 low tt ts tr splts o not nr ourt. 32
Fur 5.9: T our trs ov sr only splts A 1, A 2, n A 3 Fur 5.10: T v trs ov sr only splts A 1, A 2, n A 3 Sus 6.3. W v t ollown tr splts: A 1 = A 2 = A 3 = W n s rom Fur 5.11 low tt ts tr splts o not nr ourt. 33
Fur 5.11: T our trs ov sr only splts A 1, A 2, n A 3 Lmm 5.4. I L(A) = 9, tr r no nrns tt n m. Proo. Wn w onsr tr qurtt splts wt totl o 9 stnt l lls, w s tt t posslts r v nto 5 ss pnn on t ntrstons o t ll sts o o t tr qurtt prs. W v splts A 1, A 2, n A 3 n w tror v ( 3 2) = 3 ntrstons twn tm. For onvnn, w lwys lt A 1 =. W n ssum wtout loss o nrlty tt L(A 1 ) L(A 2 ) L(A 1 ) L(A 3 ). I ts s not t s, w n smply rll A 2 n A 3 so tt ts s tru. W lso not tt L(A 1 ) L(A 2 ) {0, 1, 2, 3} sn two qurtt splts on t sm 4 lvs r tr t sm splt or ty r momptl. T our ss o our proo wll s on ntrston trpls w wll orr trpls wr t oornts r t szs o t ntrstons twn t tr qurtt splts. T rst oornt o t ntrston trpl wll L(A 1 ) L(A 2 ), t son L(A 1 ) L(A 3 ), n t tr oornt wll L(A 2 ) L(A 3 ). Cs 1. Frst ssum tt L(A 1 ) L(A 2 ) s s lr s possl, nmly 3. Sn tr must xtly 9 stnt lvs, n only 5 r omn rom splts A 1 n A 2, w s A 3 must onsst o 4 prvously unus lls. Tror, w v t 34
ntrston trpl (3,0,0). Not tt y rlln n usn our ov ssumpton tt L(A 1 ) L(A 2 ) s lrst ntrston tt (3,0,0) s quvlnt to (0,3,0) n (0,0,3). Ts quvln rtly rus t numr o suss w must k. So w r onsrn sts wr splts A 1 n A 2 sr tr lvs w, wtout loss o nrlty, w wll nm,,. W now v two suss to onsr. Sus 1.1. W v t ollown tr splts A 1 = A 2 = A 3 = Not tt splts A 1 n A 2 mply t tr splt rom Torm 4.5. Sn ts nrn s m y only two o our splts t s not onsr trnry nrn. W n s rom Fur 5.12 low tt ts tr splts yl no trnry nrns. Fur 5.12: T two trs ov sr only splts A 1, A 2, A 3 n t ormnton nry nrn Sus 1.2. W v t ollown tr splts A 1 = A 2 = A 3 = Not tt splts A 1 n A 2 mply t splts,, n rom Torm 4.5. Sn ts nrns r m y only two o our splts t r not onsr trnry nrns. T tr n Fur 5.13 low s t only wy to sply ot A 1 n A 2 s oun urn t proo o t Bnry Inrns Torm. Sn t lvs,,, n tt rtrrly to ts tr so lon s splt A 3 s rlz, w s tt tr n no nrns rwn otr tn t tr mnton prvously. 35
Fur 5.13: Ts tr sows tt A 1, A 2, A 3 n yl no trnry nrns Cs 2. Now w onsr t s wn t ntrston trpl s (2,1,1). From t son oornt w n s tt L(A 1 ) L(A 3 ) = 1. Sn w ssum tt A 1 s lwys, w n lt ts snl lmnt o t ntrston. W now v our lvs rom A 1 plus two tonl lvs rom A 2, w mns tt w must t tr unus lvs rom t rmnr o A 3. From ts w s tt t must tt L(A 2 ). W know tt A 2 must v two lvs n ommon wt A 1 n tt on o ts s to. W n s tt ts ponts us towr two possl omntons,, n,. From ts normton, w rrv t t our suss low. Sus 2.1. W v t ollown tr splts A 1 = A 2 = A 3 = Not tt splts A 1 n A 2 mply t tr splt rom Torm 4.5. Sn ts nrn s m y only two o our splts t s not onsr trnry nrn. W n s rom Fur 5.14 low tt ts tr splts yl no trnry nrns. Fur 5.14: T trs ov sr only splts A 1, A 2, A 3 36
Sus 2.2. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.15 low tt ts tr splts yl no trnry nrns. Fur 5.15: T trs ov sr only splts A 1, A 2, A 3 Sus 2.3. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.16 low tt ts tr splts yl no trnry nrns. Sus 2.4. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.17 low tt ts tr splts yl no trnry nrns. 37
Fur 5.16: T trs ov sr only splts A 1, A 2, A 3 Fur 5.17: T trs ov sr only splts A 1, A 2, A 3 Cs 3. Now w onsr t s wn t ntrston trpl s (2,1,0). Sn t rst oornt tlls us tt L(A 1 ) L(A 2 ) = 2 w n sy tt ts ntrston must tr {, } or {, }. Suppos tt A 2 =. Tn n orr to onorm to t ntrston trpl, A 3 must tr or. Lookn t t symmtry o A 1 ( = ) w s tt ts two splts r ssntlly t sm. Usn ts prnpls, w rrv t t ollown our suss. 38
Sus 3.1. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.18 low tt ts tr splts yl no trnry nrns. Fur 5.18: T trs ov sr only splts A 1, A 2, A 3 39
Sus 3.2. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.19 low tt ts tr splts yl no trnry nrns. Fur 5.19: T trs ov sr only A 1, A 2, A 3 Sus 3.3. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.20 low tt ts tr splts yl no trnry nrns. Sus 3.4. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.21 low tt ts tr splts yl no trnry nrns. 40
Fur 5.20: T trs ov sr only splts A 1, A 2, A 3 Fur 5.21: T trs ov sr only splts A 1, A 2, A 3 Cs 4. Now w onsr t s wn t ntrston trpl s (1,1,1). For t rst oornt, w n st A 1 A 2 = {} wtout loss o nrlty. So n s w v A 1 = n A 2 =. T only vrty oms rom A 3. It must v on l n ommon wt o A 1 n A 2 ut ts l nnot. I wr n 41
lmnt o L(A 3 ), tn t woul v t orm, ut ts vs us mor tn 9 lvs. So w s tt L(A 3 ) must ontn xtly on o t ollown prs n s: {, } {, } {, } {, } Howvr, w onsr ts our posslts losly w wll not tt tr ssntlly s no rn twn t pr {.} n t pr {, }. Tus w v tr os tt l to 6 suss n t proo o our lmm. Sus 4.1. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.22 low tt ts tr splts yl no trnry nrns. Fur 5.22: T trs ov sr only splts A 1, A 2, A 3 42
Sus 4.2. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.23 low tt ts tr splts yl no trnry nrns. Fur 5.23: T trs ov sr only splts A 1, A 2, A 3 Sus 4.3. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.24 low tt ts tr splts yl no trnry nrns. Sus 4.4. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.25 low tt ts tr splts yl no trnry nrns. 43
Fur 5.24: T trs ov sr only splts A 1, A 2.A 3 Fur 5.25: T trs ov sr only splts A 1, A 2, A 3 Sus 4.5. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.26 low tt ts tr splts yl no trnry nrns. 44
Fur 5.26: T trs ov sr only splts A 1, A 2, A 3 Sus 4.6. W v t ollown tr splts A 1 = A 2 = A 3 = W n s rom Fur 5.27 low tt ts tr splts yl no trnry nrns. Fur 5.27: T trs ov sr only splts A 1, A 2, A 3 45
Lmm 5.5. I L(A) = 8, tr r no nrns tt n m. Proo. Wn w onsr tr qurtt splts wt totl o 8 stnt l lls, w s tt t posslts r v nto 6 ss pnn on t ntrstons o t ll sts o o t tr qurtt prs. An or onvnn, w lwys lt A 1 =. W n ssum wtout loss o nrlty tt L(A 1 ) L(A 2 ) L(A 1 ) L(A 3 ). I ts s not t s, w n smply rll A 2 n A 3 so tt ts s tru. W lso not tt L(A 1 ) L(A 2 ) {0, 1, 2, 3} sn two qurtt splts on t sm 4 lvs r tr t sm splt or ty r momptl. T sx ss o our proo wll n s on ntrston trpls. Cs 1. Frst w onsr t ntrston trpl (3,1,1). Ts snro rsults n two possl suss. Sus 1.1. W v t ollown tr splts A 1 = A 2 = A 3 =. Frst, not tt A 1 n A 2 mply t splt. W n s rom Fur 5.28 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.28: T trs ov sr only splts A 1, A 2, A 3 46
Sus 1.2. W v t ollown tr splts A 1 = A 2 = A 3 =. Frst, not tt A 1 n A 2 mply t splts n. W n s rom Fur 5.29 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.29: T trs ov sr only splts A 1, A 2, A 3 Cs 2. Now w onsr t ntrston trpl (3,1,0). Ts snro rsults n two possl suss. Sus 2.1. W v t ollown tr splts A 1 = A 2 = A 3 =. Frst, not tt A 1 n A 2 mply t splt. W n s rom Fur 5.30 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.30: T trs ov sr only splts A 1, A 2, A 3 47
Sus 2.2. W v t ollown tr splts A 1 = A 2 = A 3 =. Frst, not tt A 1 n A 2 mply t splts, n. W n s rom Fur 5.31 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.31: T trs ov sr only splts A 1, A 2, A 3 Cs 3. Now w onsr t ntrston trpl (2,2,2). Ts snro rsults n sx possl suss. Sus 3.1. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.32 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.32: T trs ov sr only splts A 1, A 2, A 3 48
Sus 3.2. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.33 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.33: T trs ov sr only splts A 1, A 2, A 3 Sus 3.3. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.34 low owvr, tt ts splts o not yl ny trnry nrns. Sus 3.4. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.35 low owvr, tt ts splts o not yl ny trnry nrns. 49
Fur 5.34: T trs ov sr only splts A 1, A 2, A 3 Fur 5.35: T trs ov sr only splts A 1, A 2, A 3 Sus 3.5. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.36 low owvr, tt ts splts o not yl ny trnry nrns. 50
Fur 5.36: T trs ov sr only splts A 1, A 2, A 3 Sus 3.6. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.37 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.37: T trs ov sr only splts A 1, A 2, A 3 51
Cs 4. Now w onsr t ntrston trpl (2,2,1). Ts snro rsults n svn possl suss. Sus 4.1. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.38 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.38: T trs ov sr only splts A 1, A 2, A 3 Sus 4.2. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.39 low owvr, tt ts splts o not yl ny trnry nrns. Sus 4.3. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.40 low owvr, tt ts splts o not yl ny trnry nrns. 52
Fur 5.39: T trs ov sr only splts A 1, A 2, A 3 Fur 5.40: T trs ov sr only splts A 1, A 2, A 3 Sus 4.4. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.41 low owvr, tt ts splts o not yl ny trnry nrns. 53
Fur 5.41: T trs ov sr only splts A 1, A 2, A 3 Sus 4.5. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.42 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.42: T trs ov sr only splts A 1, A 2, A 3 54
Sus 4.6. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.43 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.43: T trs ov sr only splts A 1, A 2, A 3 Sus 4.7. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.44 low owvr, tt ts splts o not yl ny trnry nrns. 55
Fur 5.44: T trs ov sr only splts A 1, A 2, A 3 Cs 5. Now w onsr t ntrston trpl (2,2,0). Ts snro rsults n sx possl suss. Sus 5.1. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.45 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.45: T trs ov sr only splts A 1, A 2, A 3 56
Sus 5.2. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.46 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.46: T trs ov sr only splts A 1, A 2, A 3 Sus 5.3. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.47 low owvr, tt ts splts o not yl ny trnry nrns. Sus 5.4. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.48 low owvr, tt ts splts o not yl ny trnry nrns. 57
Fur 5.47: T trs ov sr only splts A 1, A 2, A 3 Fur 5.48: T trs ov sr only splts A 1, A 2, A 3 Sus 5.5. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.49 low owvr, tt ts splts o not yl ny trnry nrns. 58
Fur 5.49: T trs ov sr only splts A 1, A 2, A 3 Sus 5.6. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.50 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.50: T trs ov sr only splts A 1, A 2, A 3 59
Cs 6. Now w onsr t ntrston trpl (2,1,1). Ts snro rsults n svn possl suss. Sus 6.1. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.51 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.51: T trs ov sr only splts A 1, A 2, A 3 Sus 6.2. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.52 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.52: T trs ov sr only splts A 1, A 2, A 3 60
Sus 6.3. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.53 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.53: T trs ov sr only splts A 1, A 2, A 3 Sus 6.4. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.54 low owvr, tt ts splts o not yl ny trnry nrns. Sus 6.5. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.55 low owvr, tt ts splts o not yl ny trnry nrns. 61
Fur 5.54: T trs ov sr only splts A 1, A 2, A 3 Fur 5.55: T trs ov sr only splts A 1, A 2, A 3 Sus 6.6. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.56 low, tt ts splts o not yl ny nrns. 62
Fur 5.56: T trs ov sr only splts A 1, A 2, A 3 Sus 6.7. W v t ollown tr splts A 1 = A 2 = A 3 =. W n s rom Fur 5.57 low owvr, tt ts splts o not yl ny trnry nrns. Fur 5.57: T trs ov sr only splts A 1, A 2, A 3 63
W n s tt tr o xst trnry nluns rom t xmpl low. Exmpl 5.6. Consr t qurtt splts A 1 = A 2 = A 3 = W rr k to Fur 4.8 rom Cptr 4, rprou low. 1 6 2 3 4 5 7 W n to on t two nw lvs n to ts tr so tt splts A 2 n A 3 r rlz. W s rom A 2 tt ntr nor n pl t postons 1,2, or 3. Also, rom A 3 w s tt n nnot nt to t sm vrtx. It s lr owvr tt tr r postons tt wll llow ll tr splts to sply y t sm tr: or xmpl, t poston 6 n t poston 7. W n s tt ny vl postonn o n t postons 4,5,6, or 7 wll n nr t nw qurtt. Ts nw splt, lon wt t ornl tr, tn mpls otr splts usn t nry nrn ruls rom Torm 4.5. 64
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