Math 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes.

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1 Nm: UCA ID Numr: Stion lttr: th 61 : Disrt Struturs Finl Exm Instrutor: Ciprin nolsu You hv 180 minuts. No ooks, nots or lultors r llow. Do not us your own srth ppr.

2 1. (2 points h) Tru/Fls: Cirl th right nswrs. You o NOT n to justify your nswrs. Thr xists funtion from th mpty st to th st {1}. T Thr xists funtion from th st {1} to th mpty st. F Any surjtiv funtion f : {1,2,...,100} {1,2,...,100} is ijtiv. T W wnt to slt tm of 7 (unorr) plyrs out of 60 popl. This n on in xtly 60! 53! wys. F Whn w xpn th xprssion (x y +z) 10, th offiint of x 2 y 4 z 4 is 10! 2!4!4!. T t G th grph with on vrtx n no gs. Thn G is iprtit. T Thr xists grph with 9 vrtis, ll of gr 3. F Th Sltion Sort lgorithm sorts n ojts in tim Θ(nlgn). F lg(2 n +3 n ) = Θ(n). T lg(2 n +(n!) 2 ) = Θ(nlgn). T

3 2. (5 points h) Writ own th nswr to h qustion. You o NOT n to justify your nswrs. Th numr of funtions f : {1,...,10} {1,...,15} tht r NOT injtiv is: ! 5! Dtrmin th numr of rltions on th st {1,2,...,n} tht r ithr symmtri, or rflxiv, or oth: 2 n2 n +2 n(n+1)/2 2 n(n 1)/2 In how mny wys n w istriut 100 intil ookis to 10 popl, suh tht th kth prson gts t lst k ookis, for ll k? (In othr wors, prson 1 gts t lst on ooki, prson 2 gts t lst two ookis,..., prson 10 gts t lst tn ookis.) mns 10 k=1 10 k=1 k = 100 with k = k k 0 k = 100 ( ) = 100 (10 11)/2 = = 45 so th numr is C( ,10 1) = C(54,9)

4 Clult th sum: C(51,0)+C(51,1)+C(51,2)+ +C(51,24)+C(51,25). W know tht C(51,k) = C(51,51 k). Thrfor, C(51,0)+ +C(51,25) = C(51,26)+ +C(51,51) = 1 2 (C(51,0)+ +C(51,51)) = = Th trs (with th roots rwn s th iggr ots) r: (A) isomorphi s oth fr trs n root trs; (B) isomorphi s fr trs, ut not s root trs; (C) isomorphi s root trs, ut not s fr trs; (D) not isomorphi s fr trs, n not isomorphi s root trs.

5 3. (10 points) t X {1,2,...,100} sust onsisting of 57 numrs. Show tht w n fin X suh tht +13 X. Fully justify your nswr. Consir th lmnts of X n X = {+13 X}. Togthr thy r 57 2 = 114 lmnts from 1 to 113. By th pigonhol prinipl, two r qul.

6 4. A simpl, onnt, plnr grph G hs h vrtx of gr 4. () (6 points) Show tht G hs t lst 8 fs (inluing th outr f). Fully justify your nswr. Th sum of th grs is 4v = 2, so = 2v. Eulr s formul givs v + f = 2, so v 2v+f = 2, n hn v = f 2 n = 2(f 2). Sin h f hs t lst 3 gs, w hv 3f 2 = 4(f 2), whih implis f 8. () (4 points) Giv nxmpl of suh grphg, with h vrtx of gr 4 n xtly 8 fs. Not tht in orr to hv qulity in th rgumnt ov, h f must hv xtly 3 gs. Also, thr must 12 gs n 6 vrtis. Hr s n xmpl:

7 5. (5 points h) Dtrmin if th following grphs r plnr. If th grph is plnr, rrw it so tht th gs o not ross. If th grph is not plnr, xplin why. () This is plnr, it n rrwn s:

8 () f g This is not plnr, s it ontins grph homomorphi to K 3,3, with th iprtition ing,, vrsus,,f: f g

9 6. Consir th grph from th prvious prolm, with th orr of th vrtis ing f g. f g () (5 points) Fin spnning tr using th rth-first lgorithm. Show th intrmit stps in fining th tr, y writing own th g t h stp. W strt t, thn th following gs in orr: with th rsult ing th spnning tr (,),(,),(,),(,f),(,g),(,) f g

10 () (5 points) Fin spnning tr using th pth-first lgorithm. Show th intrmit stps in fining th tr, y writing own th g t h stp. f g W strt t, thn th following gs in orr: (,),(,),(,),(,),(,f),(,g) with th rsult ing th spnning tr f g

11 () (5 points) W put wights on th gs of th grph s follows: f g Fin miniml spnning tr using Prim s lgorithm. Show th intrmit stps y writing own th g t h stp. W strt t, thn th following gs in orr: (,f),(,g),(g,),(,),(,),(,) with th rsult ing th spnning tr in olf: f g

12 7. W fin trnry tr to root tr with th following proprtis: Eh vrtx hs t most thr hilrn; Eh hil of vrtx is ll y (lft), (mil) or (right), n no two hilrn of th sm vrtx hv th sm ll. An xmpl is shown low: W fin n isomorphism of trnry trs to ijtion twn thir sts of vrtis with th following proprtis: It prsrvs th jny rltion; It tks th root to th root; It prsrvs th lls: it tks hilrn to hilrn, hilrn to hilrn, n hilrn to hilrn. For xmpl, ths r ll nonisomorphi trnry trs with 2 vrtis: (Continu on nxt pg)

13 () (3 points) Drw ll nonisomorphi trnry trs with 3 vrtis.

14 () (3 points) t n th numr of nonisomorphi trnry trs with n vrtis. Fin rurrn rltion for n. A trnry grph with n vrtis is ompos of its root n thr sutrs oming from th root: lft, mil, n right. If i,j,k not th numr of vrtis in th lft, mil, n right sutrs, rsptivly, thn i+j+k = n 1. Eh sutr is trnry grph itslf. Thrfor, n = i+j+k=n 1 i j k.

15 () (4 points) t h th hight of trnry tr with n vrtis. Show tht: h log 3 (2n+1) 1. At lvl 0 w hv t most 1 vrtx, t lvl 1 t most 3 vrtis, t lvl 2 t most 3 2 vrtis,..., t lvl h t most 3 h vrtis. Thus, th totl numr of vrtis n is t most Th inqulity n rwrittn s or h = 3h n (3 h+1 1)/2 log 3 (2n+1) h+1 h log 3 (2n+1) 1. Sin h is n intgr, w n us th iling of log 3 (2n+1).

16 Do not writ on this pg. 1 out of 20 points 2 out of 25 points 3 out of 10 points 4 out of 10 points 5 out of 10 points 6 out of 15 points 7 out of 10 points Totl out of 100 points

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