MCS100. One can begin to reason only when a clear picture has been formed in the imagination.

Transcription

1 642 ptr 10 Grps n Trs 46. Imin tt t irmsown low is mp wit ountris ll. Is it possil to olor t mp wit only tr olors so tt no two jnt ountris v t sm olor? To nswr tis qustion, rw n nlyz rp in wi ountry is rprsnt y vrtx n two vrtis r onnt y n i, n only i, t ountris sr ommon orr. 47. In tis xris rp is us to lp solv sulin prolm. Twlv ultymmrs in mtmtis prtmnt srv on t ollowin ommitts: Unrrut Eution: Tnnr, Ptrson, Ksin, on Grut Eution: Gtto, Yn, on, toiu olloquium: Sin, MMurry,s Lirry: ortzn, Tnnr, Sin irin: Gtto, MMurry, Yn, Ptrson Prsonnl: Yn, Wn, ortzn T ommitts mustllmt urin t irst wk o lsss, ut tr r only tr tim slots vill. Fin sul tt will llow ll ulty mmrs to ttn t mtins o ll ommitts on wi ty srv. To o tis, rprsnt ommitt s t vrtx o rp, n rw n twn two vrtis i t two ommitts v ommon mmr. Fin wy to olor t vrtis usin only tr olors so tt no two ommitts v t sm olor, n xplin ow to us t rsult to sul t mtins. 48. prtmnt wnts to sul inl xms so tt no stunt smor tn on xm on ny ivn y. T vrtis o t rp low sow t ourss tt r in tkn y mor tn on stunt, wit n onntin two vrtis i tr is stunt in ot ourss. Fin wy to olor t vrtis o t rp wit only our olors so tt no two jnt vrtis v t sm olor n xplin ow to us t rsult to sul t inl xms. MS100 MS135 MS101 MS130 MS102 MS120 MS110 nswrs or Tst Yoursl 1. init, nonmpty st o vrtis; init st o s; on or two vrtis ll its npoints 2. n wit sinl npoint 3. ty v t sm st o npoints 4. ty r onnt y n 5. oits npoints 6. jnt 7. isolt 8. n orr pir o vrtisll its npoints 9. rp wit no loops or prlll s 10. simpl rp wit n vrtis wos st o s ontins xtly on or pir o vrtis 11. onnt y n ; ny otr vrtx in V 1 ;ny otr vrtx in V vry vrtx in is lso vrtx in G; vry in is lso n in G; vry in s t sm npoints s it s in G 13. t numr o s tt r inint on t vrtx, wit n tt is loop ount twi 14. t sum o t rs o ll t vrtis o t rp 15. qul to twi t numr o s o t rp 16. n vn numr 10.2 Trils, Pts, n iruits On n in to rson only wn lr pitur s n orm in t imintion. W.W.Swyr, Mtmtiin s lit, 1943 T sujt o rp tory n in t yr 1736 wn t rt mtmtiin Lonr Eulr pulis ppr ivin t solution to t ollowin puzzl: T town o Könisr in Prussi (now Klininr in Russi) ws uilt t point wr two rns o t Prl Rivr m totr. It onsist o n isln n som ln lon t rivr nks.ts wr onnt ysvn ris ssown in Fiur T qustion is tis:is it possil or prson to tk wlk roun town, strtin n nin t t smlotion nrossin o t svn ris xtly on? In is oriinl ppr, Eulr i not rquir t wlk to strt n n t t sm point. T nlysis o t prolm is simplii, owvr, y in tis onition. Ltr in t stion, w isuss wlks tt strt n n t irnt points. opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

2 10.2 Trils, Pts, n iruits 643 Prl Rivr Mrin-Ern Fiur T Svn ris o Könisr To solv tis puzzl, Eulr trnslt it into rp tory prolm. noti tt ll points o ivn lnmss n intii wit otr sin prson n trvl rom ny on point to ny otr point o t sm lnmss witout rossin ri. Tus or t purpososolvin t puzzl, t mp o Könisr n intii wit t rp sown in Fiur , in wi t vrtis,,, n rprsnt ln msss n t svn s rprsnt t svn ris. Lonr Eulr ( ) ttmnn/oris Fiur Grp Vrsion o Könisr Mp In trms o tis rp, t qustion oms t ollowin: Equivlntly: Is it possil to in rout trou t rp tt strts n ns t som vrtx, on o,,, or,n trvrss xtly on? Is it possil to tr tis rp, strtin n nin t t sm point, witout vr litin your pnil rom t ppr? Tk w minuts to tink out t qustion yoursl. n you in rout tt mts t rquirmnts? Try it! Lookin or rout is rustrtin us you ontinully in yoursl t vrtx tt os not v n unus on wi to lv, wil lswr tr r unus s tt must still trvrs. I you strt t vrtx, or xmpl, tim you pss trou vrtx,, or, you us up twos us you rriv on on n prt on irnt on. So, i it is possil to in rout tt uss ll t s o t rp n strts n ns t, tn t totl numr o rrivls n prturs rom vrtx,, n must multipl o 2. Or, in otr wors, t rs o opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

3 644 ptr 10 Grps n Trs t vrtis,, n must vn. ut ty r not: () = 5, () = 3, n () = 3. n tr is no rout tt solvs t puzzl y strtin n nin t. Similr rsonin n us to sow tt tr r no routs tt solv t puzzl y strtin n nin t,, or. Tror, it is impossil to trvl ll roun t ity rossin ri xtly on. initions Trvl in rp is omplis ymovin rom on vrtx to notr lon squn o jnt s. In t rp low, or instn, you n o rom u 1 to u 4 y tkin 1 to u 2 n tn 7 to u 4.Tis is rprsnt y writin u 1 1 u 2 7 u 4. u u 3 u 1 u 4 5 u5 6 Or you oul tk t rounout rout u 1 1 u 2 3 u 3 4 u 2 3 u 3 5 u 4 6 u 4 7 u 2 3 u 3 5 u 4. rtin typs o squns o jnt vrtis n s r o spil importn in rp tory: tos tt o not v rpt, tos tt o not v rpt vrtx, n tos tt strt n n t t sm vrtx. inition Lt G rp, n lt v n w vrtis in G. wlk rom v tow is init ltrntin squn o jnt vrtis n s o G. Tus wlk s t orm v v n 1 n v n, wr t v s rprsnt vrtis, t s rprsnt s, v 0 = v, v n = w, n or ll i = 1, 2,...n,v i 1 n v i r t npoints o i.ttrivil wlk rom v to v onsists o t sinl vrtx v. tril rom v to w is wlk rom v to w tt os not ontin rpt. pt rom v to w is tril tt os not ontin rpt vrtx. los wlk is wlk tt strts n ns t t sm vrtx. iruit is los wlk tt ontins t lst on n os not ontin rpt. simpl iruit is iruit tt os not v ny otr rpt vrtx xpt t irstn lst. opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

4 10.2 Trils, Pts, n iruits 645 For s o rrn, ts initions r summriz in t ollowin tl: Rpt Rpt Strts n Ens Must ontin t E? Vrtx? t Sm Point? Lst On E? Wlk llow llow llow no Tril no llow llow no Pt no no no no los wlk llow llow ys no iruit no llow ys ys Simpl iruit no irstn ys ys lst only Exmpl Nottion or Wlks Otn wlk n spiiunmiuously y ivin itr squnos or squn o vrtis. T nxt twoxmpls sow ow tis ison.. In t rp low, t nottion rrsunmiuously to t ollowin wlk: On t otr n, t nottion 1 is miuous i us to rr to wlk. It oul mn itr 1 or v In t rp o prt (), t nottion is miuous i us to rr to wlk. It oul mn 2 or 3. On t otr n, in t rp low, t nottion rrsunmiuously to t wlk Not tt i rp G os not v ny prlll s, tn ny wlk in G isuniquly trmin y itssqun o vrtis. Exmpl Wlks, Trils Pts, n iruits In t rp low, trmin wi o t ollowin wlks r trils, pts, iruits, or simpl iruits v opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

5 646 ptr 10 Grps n Trs Solution. Tis wlk s rpt vrtx ut os not v rpt, so itis tril rom to ut not pt.. Tis is just wlk rom to.itis not tril us its rpt.. Tis wlk strts n ns t, ontins t lst on, nos not v rpt, so itis iruit. Sin t vrtx is rpt in t mil, it is not simpl iruit.. Tis wlk strts n ns t, ontins t lst on, os not v rpt, n os not v rpt vrtx. Tus it is simpl iruit.. Tis is just los wlk strtin n nin t.itis not iruit us 1 is rpt.. T irst vrtx o tis wlk is t sm s its lst vrtx, ut itos not ontin n, n soitis not iruit. It is los wlk rom to.(itislso tril rom to.) us most o t mjor vlopmnts in rp tory v ppn rltivly rntly n in vrity o irnt ontxts, t trms us in t sujt v not n stnriz. For xmpl, wt tis ook lls rp is somtims ll multirp, wt tis ook lls simpl rp is somtims ll rp, wt tis ook lls vrtx is somtims ll no, n wt tis ook lls n is somtims ll n r. Similrly, inst o t wor tril, t wor pt is somtims us; inst o t wor pt, twors simpl pt r somtims us; n inst o t wors simpl iruit,twor yl issomtimsus. T trminoloy in tis ook is mon t most ommon, ut iyou onsult otr sours, sur to k tir initions. onntnss It is sy to unrstn t onpt o onntnss on n intuitiv lvl. Rouly spkin, rp is onnt i it is possil to trvl rom ny vrtx to ny otr vrtx lon squn ojnt s o t rp. T orml inition o onntnss isstt in trms o wlks. inition Lt G rp. Two vrtis v n w o G r onnt i, n only i, tr is wlk rom v to w. Trp G is onnt i, n only i, ivn ny two vrtis v n w in G, tr is wlk rom v to w. Symolilly, G is onnt vrtis v, w V (G), wlkrom v to w. I you tk t ntion o tisinition, you will s tt rp G is not onnt i, n only i, tr r two vrtis o G tt r not onnt y ny wlk. Exmpl onnt n isonnt Grps Wi o t ollowin rps r onnt? v 2 v2 v 4 v v v3 1 6 v 8 v 7 () () () opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

6 10.2 Trils, Pts, n iruits 647 Solution T rp rprsnt in () isonnt, wrs tos o () n () r not. To unrstn wy () is not onnt, rll tt in rwin o rp, two s myross t point tt is not vrtx. Tus t rp in () n rrwn s ollows: Som usul ts rltin iruits n onntnss r ollt in t ollowin lmm. Proos o () n () r lt or t xriss. T proo o ()is in Stion Lmm Lt G rp.. I G isonnt, tn ny twoistint vrtis o G n onnt y pt.. I vrtis v n w r prt o iruit in G n on is rmov rom t iruit, tn tr still xists tril rom v to w in G.. IG is onnt n G ontins iruit, tn n o t iruit n rmov witout isonntin G. Look k texmpl T rps in () n () r ot m up o tr pis, o wi is itsl onnt rp. onnt omponnt o rp is onnt surp o lrst possil siz. inition rp is onnt omponnt o rp G i, n only i, 1. issurp o G; 2. isonnt; n 3. noonnt surp o G s s surp n ontins vrtis or s tt r not in. T t is tt ny rp is kin o union o itsonnt omponnts. Exmpl onnt omponnts Fin ll onnt omponnts o t ollowin rp G v 8 v 7 opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

7 648 ptr 10 Grps n Trs Solution G s tr onnt omponnts: 1, 2,n 3 wit vrtx sts V 1, V 2,n V 3 n sts E 1, E 2,n E 3, wr Eulr iruits V 1 ={,, }, E 1 ={ 1, 2 }, V 2 ={ }, E 2 =, V 3 ={,,v 7,v 8 }, E 3 ={ 3, 4, 5 }. Now w rturn to onsir nrl prolms similr to t puzzl o t Könisr ris. T ollowin inition ism in onor o Eulr. inition Lt G rp. n Eulr iruit or G is iruit tt ontins vry vrtx n vry o G. Tt is, neulr iruit or G is squn ojnt vrtis n s in G tt s t lst on, strts n ns t t sm vrtx, uss vry vrtx o G t lst on, n uss vry o G xtly on. T nlysisus rlir to solv t puzzl o t Könisr ris nrlizs to prov t ollowin torm: Torm I rp s n Eulr iruit, tn vry vrtx o t rp s positiv vn r. Proo: Suppos G is rp tt s n Eulr iruit. [W must sow tt ivn ny vrtx v o G, t r o v is vn.] Lt v ny prtiulr ut ritrrily osn vrtx o G.Sin t Eulr iruit ontins vry o G,itontins ll s inint on v. Nowimin tkin journy tt ins in t mil o on o t s jnt to t strt o t Eulr iruit nontinus roun t Eulr iruit to n in t mil o t strtin. (S Fiur Tr is sustrtin us teulr iruit s t lst on.) E tim v is ntr y trvlin lon on, it is immitly xit y trvlin lon notr (sin t journy ns in t mil o n ). Strt r v 0 First ntry/xit pir o s In tis xmpl, t Eulr iruit is v 0 v 0,n v is. E tim is ntr y on, it is xit y notr. Son ntry/xit pir o s Fiur Exmpl or t Proo o Torm opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

8 10.2 Trils, Pts, n iruits 649 usteulr iruit uss vry o G xtly on, vry inint on v is trvrs xtly on intis pross. n t s inint on v our in ntry/xit pirs, n onsquntly t r o v must positiv multipl o 2. ut tt mns tt v s positiv vn r [s ws to sown]. Rll tt t ontrpositiv o sttmnt is loilly quivlnt to t sttmnt. T ontrpositiv o Torm is s ollows: ontrpositiv Vrsion o Torm I som vrtx o rp s o r, tn t rp os not v n Eulr iruit. Tis vrsion o Torm isusul or sowin tt ivn rp os not v n Eulr iruit. Exmpl Sowin Tt Grp os Not v n Eulr iruit Sow tt t rp low os not v n Eulr iruit Solution Vrtis n ot v r 3, wiiso. ny (t ontrpositiv orm o) Torm , tis rp os not v n Eulr iruit. Now onsir t onvrs o Torm : I vry vrtx o rp s vn r, tn t rp s n Eulr iruit. Is tis tru? T nswr is no. Tr is rp G su tt vry vrtx o G s vn r ut G os not v n Eulr iruit. In t, tr r mny su rps. T illustrtion low sows on xmpl Evry vrtx s vn r, ut t rp os not v n Eulr iruit. Not tt t rp in t prin rwin is not onnt. It turns out tt ltou t onvrs o Torm is ls, moii onvrs is tru: I vry vrtx o rp s positiv vn r n i t rp is onnt, tn t rp s n Eulr iruit. T proo o tis t isonstrutiv: It ontins n loritm to in n Eulr iruit or ny onnt rp in wi vry vrtx s vn r. opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

9 650 ptr 10 Grps n Trs Torm I rp G is onnt n t r o vry vrtx o G is positiv vn intr, tn G s n Eulr iruit. Proo: Suppos tt G is ny onnt rp n suppos tt vry vrtx o G is positiv vn intr. [W must in n Eulr iruit or G.] onstrut iruit y t ollowin loritm: Stp 1: Pik ny vrtx v o G t wi tostrt. [Tis stp n omplis us t vrtx st o G is nonmpty y ssumption.] Stp 2: Pik nysqun ojnt vrtis n s, strtin n nin t v n nvr rptin n. ll t rsultin iruit. [Tis stp n prorm or t ollowin rsons: Sin t r o vrtx o G is positiv vn intr, s vrtx o G is ntr y trvlin on on, itr t vrtx is v itsl n tr is no otr unus jnt to v, or t vrtx n xit y trvlin on notr prviously unus. Sin t numr o s o t rp is init (y inition o rp), t squn o istint s nnot o on orvr. T squn n vntully rturn to v us t r o v is positiv vn intr, n so i n onnts v to notr vrtx, tr must irnt tt onnts k to v.] Stp 3: k wtr ontins vry n vrtx o G. Iso, is n Eulr iruit, n w r inis. I not, prorm t ollowin stps. Stp 3: Rmov ll s o rom G n lso ny vrtis tt om isolt wn t s o r rmov. ll t rsultin surp G. [Not tt G my not onnt (s illustrt in Fiur ), ut vry vrtx o G s positiv, vn r (sin rmovin t s o rmovs n vn numr o s rom vrtx, t irn o two vn intrs is vn, n isolt vrtis wit r 0 wr rmov.)] u v w G: G' Fiur Stp 3: Pik ny vrtx w ommon to ot n G. [Tr must t lst on su vrtx sin G is onnt. (S xris 44.) (In Fiur tr r two su vrtis: u n w.)] Stp 3: Pik nysqun ojnt vrtis n s o G, strtin n nin t w n nvr rptin n. ll t rsultin iruit. [Tis n on sin vrtx o G s positiv, vn r n G is init. S t justiition or stp 2.] opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

10 10.2 Trils, Pts, n iruits 651 Stp 3: Pt n totr to rt nw iruit s ollows: Strt t v n ollow ll t wy to w. Tn ollow ll t wy ktow. tr tt, ontinu lon t untrvl portion o to rturn to v. [T t o xutin stps 3 n 3 or t rp o Fiur is sown in Fiur ] ' u v w G: Fiur '' Stp 3: Lt = n o k tostp 3. Sin t rp G is init, xution o t stps outlin in tis loritm must vntully trmint. t tt point n Eulr iruit or G will v n onstrut. (Not tt us o t lmnt o oi instps 1, 2, 3, n 3, vrity o irnt Eulr iruits n prou yusin tis loritm.) Exmpl Finin n Eulr iruit Us Torm to k tt t rp low s n Eulr iruit. Tn us t loritm rom t proo o t torm to in n Eulr iruit or t rp. i Solution j Osrv tt () = () = () = ( ) = () = (i) = ( j) = 2 n tt () = () = () = 4. n ll vrtis v vn r. lso, t rp is onnt. Tus, y Torm , t rp s n Eulr iruit. To onstrutneulr iruit usin t loritm o Torm , lt v = n lt :. is rprsnt y t ll s sown low j i opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

11 652 ptr 10 Grps n Trs Osrv tt is not n Eulr iruit or t rp ut tt intrsts t rst o t rp t. Lt Pt into to otin : ji. : ji. St =. Tn is rprsnt y t ll s sown low i j Osrv tt is not n Eulr iruit or t rp ut tt it intrsts t rst o t rp t. Lt Pt into to otin :. : ji. St =. Tn is rprsnt y t ll s sown low i 11 j Sin inlus vry o t rp xtly on, is n Eulr iruit or t rp. In xris 45 t t n o tis stion you r sk to sow tt ny rp wit n Eulr iruit is onnt. Tis rsult n omin wit Torms n to iv omplt rtriztion o rps tt v Eulr iruits, s stt in Torm Torm rp G s n Eulr iruit i, n only i, G isonnt n vry vrtx o G s positiv vn r. orollry to Torm ivs ritrion or trminin wn it is possil to in wlk rom on vrtx o rp to notr, pssin trou vry vrtx o t rp t lst on n vry o t rp xtly on. inition Lt G rp, n lt v n w two istint vrtis o G.nEulr tril rom v to w is squnojnt s n vrtis tt strts t v,ns t w,psss trou vry vrtx o G t lst on, n trvrss vry o G xtly on. opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

12 10.2 Trils, Pts, n iruits 653 orollry Lt G rp, n lt v n w two istint vrtis o G. Tr is n Eulr pt rom v to w i, n only i, G is onnt, v n w v o r, n ll otr vrtis o G v positiv vn r. T proo o tisorollry is lt s n xris. Exmpl Finin n Eulr Tril T loor pln sown low is or ous tt is opn or puli viwin. Is it possil to in tril tt strts in room, ns in room, n psss trou vry intrior oorwy o t ous xtly on? I so, in su tril. F G I E J K Solution Lt t loor pln o t ous rprsnt y t rp low. G I F K E J E vrtx o tis rp s vn r xpt or n, o wisr 1. n y orollry , tr is n Eulr pt rom to. Onsu tril is G FEI EK J. Sir Wm. milton ( ) ttmnn/oris miltonin iruits Torm ompltly nswrs t ollowin qustion: Givn rp G, is it possil to in iruit or G in wi ll t s o G ppr xtly on? rlt qustion is tis: Givn rp G, is it possil to in iruit or G in wi ll t vrtis o G (xpt t irstn t lst) ppr xtly on? In 1859 t Iris mtmtiin Sir Willim Rown milton introu puzzl in t sp o oron (O-k--EE-ron). (Fiur ontins rwin o oron, wi is soliiur wit 12 intil pntonl s.) Fiur oron opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

13 654 ptr 10 Grps n Trs E vrtx ws ll wit t nm o ity Lonon, Pris, on Kon, Nw York, n so on. T prolm milton pos ws to strt t on ity n tour t worl y visitin otr ity xtly on n rturnin to t strtin ity. On wy to solv t puzzl is to imin t sur o t oron strt out n li lt in t pln, s ollows: T iruit not wit lk lins is on solution. Not tt ltou vry ity is visit, mny s r omitt rom t iruit. (Mor iiult vrsions o t puzzl rquir tt rtin itis visit in rtin orr.) T ollowin inition ism in onor o milton. inition Givn rp G, miltonin iruit or G is simpl iruit tt inlus vry vrtx o G. Tt is, miltonin iruit or G is squn o jnt vrtis n istint s in wi vry vrtx o G pprs xtly on, xpt or t irstn t lst, wi r t sm. Not tt ltou n Eulr iruit or rp G must inlu vry vrtx o G, it my visit som vrtis mor tn on n n my not miltonin iruit. On t otr n, miltonin iruit or G os not n to inlu ll t s o G n n my not n Eulr iruit. spit t nloous-sounin initions o Eulr n miltonin iruits, t mtmtis o t two r vry irnt. Torm ivs simpl ritrion or trminin wtr ivn rp s n Eulr iruit. Unortuntly, tr is no nloous ritrion or trminin wtr ivn rp s miltonin iruit, nor is tr vn n iint loritm or inin su iruit. Tr is, owvr, simpl tniqu tt n us in mny ss to sow tt rp os not v miltonin iruit. Tis ollows rom t ollowin onsirtions: Suppos rp G wit t lst two vrtis s miltonin iruit ivn onrtly s : v v n 1 n v n. Sin is simpl iruit, ll t i r istint n ll t v j r istint xpt tt v 0 = v n.lt t surp o G tt is orm usin t vrtis n s o. n xmpl o su n issown low. is init y t lk lins. Not tt s t sm numr o s s it s vrtis sin ll its n s r istintnso r its n vrtis,,...,v n.lso, yinition o miltonin iruit, opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

14 10.2 Trils, Pts, n iruits 655 vry vrtx o G is vrtx o,n isonntsinny two o its vrtis li on iruit. In ition, vry vrtx o s r 2. T rson or tis is tt tr r xtly two s inint on ny vrtx. Ts r i n i+1 or ny vrtx v i xpt v 0 = v n,n ty r 1 n n or v 0 (= v n ).Ts osrvtions v stlis t trut o t ollowin proposition in ll ss wr G s t lst two vrtis. Proposition I rp G s miltonin iruit, tn G s surp wit t ollowin proprtis: 1. ontins vry vrtx o G. 2. isonnt. 3. s t smnumr o s s vrtis. 4. Evry vrtx o s r 2. Not tt i G ontins only on vrtx n G s miltonin iruit, tn t iruit s t orm v v, wr v is t vrtx o G n is n inint on v.intiss, t surp onsistin o v n stisis onitions (1) (4) o Proposition Rll tt t ontrpositiv o sttmnt is loilly quivlnt to t sttmnt. T ontrpositiv o Proposition sys tt i rp G os not v surp wit proprtis (1) (4), tn G os not v miltonin iruit. Exmpl Sowin Tt Grp os Not v miltonin iruit Prov tt t rp G sown low os not v miltonin iruit. Solution I G s miltonin iruit, tn y Proposition , G s surp tt (1) ontins vry vrtx o G,(2) isonnt, (3) s t smnumr o s s vrtis, n (4) is su tt vry vrtx s r 2. Suppos su surp xists. In otr wors, suppos tr is onnt surp o G su tt s iv vrtis (,,,, ) n iv s n su tt vry vrtx o s r 2. Sin t r o in G is 4n vry vrtx o s r 2, two s inint on must rmov rom G to rt. E {, } nnot rmov us iit wr, vrtx woul v r lss tn 2 in. Similr rsonin sows tt s {, }, {, }, n {, } nnot rmov itr. It ollows tt t r o in must 4,wi ontrits t onition tt vry vrtx in s r 2 in. n nosu surp xists, nso G os not v miltonin iruit. T nxt xmpl illustrts typ o prolm known s trvlin slsmn prolm. It is vrition o t prolm o inin miltonin iruit or rp. opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

15 656 ptr 10 Grps n Trs Exmpl Trvlin Slsmn Prolm Imin tt t rwin low is mp sowin our itis n t istns in kilomtrs twn tm.suppos tt slsmn must trvl to ity xtly on, strtin n nin in ity. Wi rout rom ity to ity will minimiz t totl istn tt must trvl? Solution Tis prolm n solv y writin ll possil miltonin iruits strtin n nin t n lultin t totl istn trvl or. Rout Totl istn (In Kilomtrs) = = = [ kwrs] 155 [ kwrs] 125 [ kwrs] Tus itr rout or ivs minimum totl istn o 125 kilomtrs. T nrl trvlin slsmn prolm involvs inin miltonin iruit to minimiz t totl istn trvl or n ritrry rp wit n vrtis in wi is mrk wit istn. On wy to solv t nrl prolm is to us t mto o Exmpl : Writ own ll miltonin iruits strtin n nin t prtiulr vrtx, omput t totl istn or, n pik on or wi tis totl is miniml. owvr, vn or mium-siz vlus o n tis mto is imprtil. For omplt rp wit 30 vrtis, tr woul (29!)/2 = miltonin iruits strtin n nin t prtiulr vrtx to k. Evn i iruit oul oun n its totl istn omput in just on nnoson, it woul rquir pproximtly yrs to inis t omputtion. t prsnt, tr is no known loritm or solvin t nrl trvlin slsmn prolm tt ismor iint. owvr, tr r iint loritms tt in prtty oo solutions tt is, iruits tt, wil not nssrily vin t lstpossil totl istns,vsmllr totl istns tn most otr miltonin iruits. Tst Yoursl 1. Lt G rp n lt v n w vrtis in G. () wlk rom v to w is. () tril rom v to w is. () pt rom v to w is. () los wlk is. () iruit is. () simpl iruit is. () trivil wlk is. () Vrtis v n w r onnt i, n only i,. opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

16 10.2 Trils, Pts, n iruits rp isonnt i, n only i,. 3. Rmovin n rom iruit in rp os not. 4. n Eulr iruit in rp is. 5. rp s n Eulr iruit i, n only i,. 6. Givn vrtis v n w in rp, tr is n Eulr pt rom v to w i, n only i,. 7. miltonin iruit in rp is. 8. I rp G s miltonin iruit, tn G s surp wit t ollowin proprtis:,,, n. 9. trvlin slsmn prolm involvs inin tt minimizs t totl istn trvl or rp in wi is mrk wit istn. Exris St In t rp low, trmin wtr t ollowin wlks r trils, pts, los wlks, iruits, simpl iruits, or just wlks.. v v In t rp low, trmin wtr t ollowin wlks r trils, pts, los wlks, iruits, simpl iruits, or just wlks v v ow mny pts r tr rom to?. ow mny trils r tr rom to?. ow mny wlks r tr rom to? 5. onsir t ollowin rp ow mny pts r tr rom to?. ow mny trils r tr rom to?. ow mny wlks r tr rom to? 6. n wos rmovl isonnts t rp o wi it is prt is ll ri. Fin ll ris or o t ollowin rps.. 5. v v v 8 v 7 3. Lt G t rp 0 v5 v7 1 v 8 v 9 2 n onsir t wlk n tis wlk writtn unmiuously s? Wy?. n tis wlk writtn unmiuously s 1 2?Wy? 4. onsir t ollowin rp. 7. Givn ny positiv intr n, () in onnt rp wit n s su tt rmovl o just on isonnts t rp; () in onnt rp wit n s tt nnot isonnt y t rmovl o ny sinl opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

17 658 ptr 10 Grps n Trs 8. Fin t numr o onnt omponnts or o t ollowin rps i 15. s t u v w r z y x. v u w z y x v 0. j i. 18. Is it possil to tk wlk roun t ity wos mp is sown low, strtin n nin t t sm point n rossin ri xtly on? I so, ow n tis on? E F 9. E o () () sris rp. In s nswr ys, no, or not nssrily to tis qustion: os t rp v n Eulr iruit? Justiyyour nswrs.. G is onnt rp wit iv vrtis o rs 2, 2, 3, 3,n 4.. G is onnt rp wit iv vrtis o rs 2, 2, 4, 4, n 6.. G is rp wit iv vrtis o rs 2, 2, 4, 4, n T solution or Exmpl sows rp or wi vry vrtx s vn r ut wi os not v n Eulr iruit. Giv notr xmpl o rp stisyin ts proprtis. 11. Is it possil or itizn o Könisr to mk tour o t ity n ross ri xtly twi? (S Fiur ) Wy? trmin wi o t rps in v Eulr iruits. I t rp os not v n Eulr iruit, xplin wy not. I it os v n Eulr iruit, sri on v 9 v 8 v 0 v3 Rivr For o t rps in 19 21, trmin wtr tr is n Eulr pt rom u to w. I tr is, in su pt. 19. v 7 w v 0 u E 20. u w v 7 opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

18 10.2 Trils, Pts, n iruits v 7 u v 0 F E G w In in miltonin iruits or tos rps tt v tm. Explin wy t otr rps o not T ollowin is loor pln o ous. Is it possil to ntr t ous in room, trvl trou vry intrior oorwy o t ous xtly on, n xit out o room E?Iso, ow n tis on? G F E 30. v v 7 Fin miltonin iruits or o t rps in 23 n v 0 v 7 l k i Giv two xmpls o rps tt v Eulr iruits ut not miltonin iruits. Giv two xmpls o rps tt v miltonin iruits ut not Eulr iruits. Giv two xmpls o rps tt v iruits tt r ot Eulr iruits n miltonin iruits. Giv two xmpls o rps tt v Eulr iruits n miltonin iruits tt r not t sm. j Sow tt non o t rps in s miltonin iruit i j opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.

19 660 ptr 10 Grps n Trs 36. trvlr in Europ wnts to visit o t itis sown on t mp xtly on, strtin n nin in russls. T istn (in kilomtrs) twn pir o itis is ivn in t tl. Fin miltonin iruit tt minimizs t totl istn trvl. (Us t mp to nrrow t possil iruits own to just w. Tn us t tl to in t totl istn or o tos.) russls üsslor Luxmour Muni Pris , rlin russls üsslor Luxmour Pris Muni rlin russls üsslor Luxmour Muni Prov tt i wlk in rp ontins rpt, tn t wlk ontins rpt vrtx.. Explin ow it ollows rom prt () tt ny wlk wit no rpt vrtx s no rpt. 38. Prov Lmm (): I G is onnt rp, tn ny two istint vrtis o G n onnt y pt. 39. Prov Lmm (): I vrtis v n w r prt o iruit in rp G n on is rmov rom t iruit, tn tr still xists tril rom v to w in G. 40. rw pitur to illustrt Lmm (): I rp G is onnt n G ontins iruit, tn n o t iruit n rmov witout isonntin G. 41. Prov tt i tr is tril in rp G rom vrtx v to vrtx w, tn tr is tril rom w to v I rp ontins iruit tt strts n ns t vrtx v, os t rp ontin simpl iruit tt strts n ns t v?wy? 43. Prov tt i tr is iruit in rp tt strts n ns t vrtx v n i w is notr vrtx in t iruit, tn tr is iruit in t rp tt strts n ns t w. 44. Lt G onnt rp, n lt ny iruit in G tt os not ontin vry vrtx o. LtG t surp otin y rmovin ll t s o rom G n lso ny vrtis tt om isolt wn t s o r rmov. Prov tt tr xists vrtx v su tt v is in ot n G. 45. Prov tt ny rp wit n Eulr iruit is onnt. 46. Prov orollry For wt vlus o n os t omplt rp K n wit n vrtis v () n Eulr iruit? () miltonin iruit? Justiyyour nswrs. For wt vlus o m n n os t omplt iprtit rp on (m, n) vrtis v () n Eulr iruit? () miltonin iruit? Justiyyour nswrs. Wt is t mximum numr o s simpl isonnt rp wit n vrtis n v? Prov your nswr. Sow tt rp is iprtit i, n only i, it os not v iruit wit n o numr o s. (S xris 37 o Stion 10.1 or t inition o iprtit rp.) nswrs or Tst Yoursl 1. () init ltrntin squn ojnt vrtis n s o G () wlk tt os not ontin rpt () tril tt os not ontin rpt vrtx () wlk tt strts n ns t t sm vrtx () los wlk tt ontins t lst on n os not ontin rpt () iruit tt os not v ny rpt vrtx otr tn t irst n t lst () wlk onsistin o sinl vrtx n no () tr is wlk rom v to w 2. ivn ny two vrtis in t rp, tr is wlk rom on to t otr 3. isonnt t rp 4. iruit tt ontins vry vrtx n vry o t rp 5. t rp is onnt, n vry vrtx s positiv, vn r 6. t rp is onnt, v n w v o r, n ll otr vrtis v positiv vn r 7. simpl iruit tt inlus vry vrtx o t rp 8. ontins vry vrtx o G; isonnt; s t sm numr o s s vrtis;vry vrtx o s r 2 9. miltonin iruit opyrit 2010 n Lrnin. ll Rits Rsrv. My not opi, snn, or uplit, in wol or in prt. u to ltroni rits, som tir prty ontnt my supprss rom t ook n/or ptr(s). Eitoril rviw s m tt ny supprss ontnt os not mtrilly t t ovrll lrnin xprin. n Lrnin rsrvs t rit to rmov itionl ontnt t ny tim i susqunt rits rstritions rquir it.