Grade 7/8 Math Circles March 4/5, Graph Theory I- Solutions

Transcription

1 ulty o Mtmtis Wtrloo, Ontrio N ntr or ution in Mtmtis n omputin r / Mt irls Mr /, 0 rp Tory - Solutions * inits lln qustion. Tr t ollowin wlks on t rp low. or on, stt wtr it is pt? ow o you know? () n () r pts sin ty o not rpt vrtis. Noti tt is rpt in () n (), n is rpt in (). () () () --- () ()

2 . in 0 wlks rom to in t ollowin rp. ow mny o ts wlks r pts? ow mny possil pts r tr twn n. (int: You my wnt to rrw t rp) 0 Tr r mny wlks possil. Stunts soul striv to in t lst 0. Rmmr tt wlks n inlu rpt vrtis. Tr r pts rom to : -, --, , --0-, , , -----, in ll t yls in t ollowin rp. Tr r yls: ----, ----, ---, -----, , Wi s soul to t rp low to mk it? Writ t nms o t s n tn rw tm on t rp. T s {,}, {,}, n {,} soul so t rp looks lik t on low.

3 j Ptrsn rp. k-rulr rp is rp wit k s inint to vrtx. n xmpl o tis is t Ptrsn rp wi is typ o -rulr rp. i u, Ryn, Vin, mily, n Nin o to prty. Wn ty t tr ty wnt to sk ns ut ty only v tim to sk two otr popl s ns. rw two irnt rps to sow ow tis n ppn. ow mny nsks r tr? o t ruls or rwin rp mk sns in tis sitution? rps will vry. vrtx soul inint to s. Tr soul s totl. Tr r nsks. Ys, t ruls mk sns sin you n not sk ns wit yoursl n it os not mk sns to v two irnt nsks twn t sm popl sin ty v lry skn ns on.. Sr s lttr tt s wnts to iv to mily. ut ty won t s otr us ty r in irnt itis. nswr t ollowin qustions. T quikst wy is t ollowin pt: Sr-Vin-Ryn-u-Nin-mily mily Sr mil Tim si Ryn Vin wrn u ss lton Nin () Wi popl must s otr in orr or Sr to t t lttr to mily? n otr wors, i ts popl on t mt tn it is impossil or Sr to t t lttr to mily. Ryn n Vin, u n Nin, n Nin n mily must mt. () Wt o ts vrtis v in ommon? Ty r ll ris. () Ryn n Vin on t mt, wt ns out t rp (otr tn rmovin n )? T rp is not onnt. Tr r now omponnts. () Vin livs r rom Sr so s osn t wnt to iv im t lttr. Wt is t stst wy now? Sr-Tim-wrn-lton-ss-Vin-Ryn-u-Nin-mily () lton s lttr tt wnts to iv to si. Wt is t stst wy or im to o tis? lton-ss-vin-ryn-mil or u-si

4 . r t ollowin rps plnr? so, rw plnr rprsnttion. not, xplin wy. int: Only on o ts r plnr. Try inin or, suivisions. () Plnr, rwin my vry () Non-plnr: suivision 0 () Non-plnr:, suivision () Non-plnr: suivision. Rrw rp to sow it is plnr. () rwin my vry. () rwin my vry. i

5 . wor rp s wors s t vrtis. Two wors r jnt i ty ir y lttr. rt onnt wor rp tt ontins t ollowin wors: lo, top, t, mu n r. nswrs my vry. r. mu. t. mu. ru. pt. tu. u. pot. pop. t. tp. o. top. lo. 0. low is mp o Sout mri. rt rp wit t vrtis rprsntin ountris n t s joinin tos ountris wi sr ln orr. So il n Pru r jnt, ut Pru n Uruuy r not. Us your rp to nswr t qustions low. olumi Vnzul uyn Surinm rn uin uor Pru rzil olivi il Pruy Uruuy ourtsy o t.om rntin () in t minimum olourin o t rp, n trmin olour or ountry. Wt out or ountris tt tt sr vry smll orr. Sin orrs o not ross, tr must wy to rw plnr rp o t mp. T -olour torm sys tt ny plnr rp is -olourl. ltou it is possil tt plnr rp is lss tn -olourl, tr is surp (mnin smllr rp witin t oriinl) o (onsistin o rzil, olivi, Pruy, n rntin). Rmmr tt nots t omplt rp wit vrtis tt r ll jnt to otr, wi rquirs t lst olours.

6 () Wn trin oos ovr ln, $00 tx is pi to ountry wi t oos trvl trou. So i olumi slls o to Vnzul, t lst mount o tx pi is $00. i. Wt is t lst mount o tx pi on wool sipp rom uor to Pruy? $00 ii. Wt ountry n tr wit t most otrs or xtly $00? rzil iii. rzil riss it s txs to $00. Wt is t pst mount o txs pi to sip Vnzuln oil to Pruy? Wt out rn uin to olivi? Vnzul to Pruy: $00; rn uin to olivi: $00 iv. To nour tr, Pru will not tx ny oos oin tou t ountry. il n Surinm wnt to tr lumr n wt. Wt rout will rsult in t lst mount o txs? ow mu will ty py? Surinm, uyn, Vnzul, olumi, Pru, il; $00 () vin wnts to visit vry ountry in Sout mri on is vtion. osn t wnt to visit ny ountry mor tn on sin rossin t orr n tk lon tim! will ly into im, Pru to in is trip n ly out o Montvio, Uruuy. n wt orr soul visit ountry? Pru, uor, olumi, Vnzul, uyn, Surinm, rn uin, rzil, Pruy, olivi, il, rntin, Uruuy. *Two rps r isomorpi i ty r t sm rp rwn in irnt wy. Tis mns tt ll t onntions r t sm, n tr r xtly t sm numr o vrtis, ut t vrtis my v irnt nms n t rps my look irnt. trmin i t ollowin pirs o rps r isomorpi. so, sow wi vrtis mt, or xplin wy not. () Not isomorpi: tr is yl o lnt in t irst, ut not in t son. () somorpi s sown (solutions my vry)

7 () somorpi s sown (solutions my vry) 0 i j. ulr s Oil is s sttion in in t ountry o Mtisun. T O o t ompny, onr, wnts to visit ll is s sttions in t ountry to nsur tt on is stok wit nou ul. owvr, osn t wnt to visit ny town mor tn on s is vry usy mn. Sust rout onr oul tk rom t oi in t pitl Mttopi, trou town xtly on, n rturn to t pitl. Not tt spit t intrstion o tr iwys in t ntr o t ountry, tr is no wy to n rom on ro to notr (i.. tr is no irt ro rom Mttopi to usston, t.) nswrs my vry. On solution is Mttopi, irl ity, rtion lls, Rius Rivr, Tims Town, qutionvill, usston, Vrtx Vill, Mttopi Vrtx Vill Tims Town usston qutionvill Rius Rivr Mttopi irl ity rtion lls