201midterm2-sol - SIMON FRASER UNIVERSITY DEPARTMENT OF...

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Unformatted text preview: SIMON FRASER UNIVERSITY DEPARTMENT OF MATHEMATICS Midterm 2 MACM 201 Spring 20103 Instructor: Dr. L. Stacho March 20, 2013, 12:30 — 1:20 p.m. Name: 6 blU 3&0“ 3> (please print) family name given name SFU ID: m student number SFU—emar'l Signature: W W m Instfiittions: 1. Do not open this booklet until told to do so. ‘2. Write your name above in block letters. Write your SFU student number and email lD on the line pro— 3. Write your answer in the space provided below the ' question. If additional space is needed then use the back of the previous page. Your final answer should be simplified as far as is reasonable. 4. Make the method you are using clear in every case unless it is explicitly stated that no explanation is needed. 5. This exam has 3 questions on 4 pages (not includ— ing this cover page). Once the exam begins please check to make sure your exam is complete. 6. No calculators. books. papers, or electronic devices shall be within the reach of a student during the examination. 7. During the examination, communicating with, or deliberately exposing written papers to the view of. other examinees is forbidden. MA CM 201 Page 1 of 4 1. Recurrence Relations [5] (a) Let A : {001.011.1111}. Now, let A” be the set of all binary strings of length n, that can be obtained by concatenating strings in A. For example. 00110111 E A8, but 10000000 e Ag. Let an 2 i.e. an counts the number of strings in A“. Find the recurrence relation for on. Do not need to solve it. Q\ : \ E o 3 as 2 ‘1 C00 i 013 QB _ S- iogo‘003‘0\0\0\1‘\\\3 " .\ I l A“ n g .D S 6 WE. ¥n3l12§€Also Dlt ‘ l‘ ’2 g ‘9‘ \ g e A “13 shim 5133‘ g‘ e A .— a c; 2 .1“ hence vali a“: Ml -t' (11m2 + (Laws ‘ Gr l, 2. i as y b Solve the recurrence relation an 2 1 4a“. 1 m 4a” * 5”, where (10 = 1 and a1 = 2. + r« l - a“: 6193a? ' “5*- ouh" 0* 61"!“ . ‘9 . 'C’tlmmcl’wrflci. quri-leo q». 5 S ‘g r T k l m "‘ “inf” ‘rurlzil 0‘“ r 9‘ a l“ I Ina-'2 m“ . . “F S“ all“; C2 + (32“)? AK : ow; flqAS QS-A:‘ZO-P\“‘1A—l h h ‘5.“ q“_C‘-l2. Tan“ "g‘ l0 ‘3q0 :: Cll ~% 3% 0': Us... 1-? 01‘ :10I +102»_5: g (3,2,ch MACJLf 201 Page 2 of 4 2. Graph Theory l [4] (a) Let n 3 6. How many subgraphs on at least 3 and at most 77 — 3 vertices does the graph Kn have? n LL be» 8.!" C195, .on lg Ucrificezg LLéIZ’B‘.-.‘V\."1>E : (k>-—w¢a) "l1: Cln S’L U [ _ pessile Vans QVn-ouxa ck‘uszm llama, ii c>ll— Suck 3°53“th l¢ UUHLLS t: M [n] 2(5) £2) , mag aucluae/ma—aucluae eolasz ‘5 Z K - . amufié WV. (964‘ 3 2:8 [3] (b) Let G be a loop—free connected undirected graph without multiple edges. Suppose G has at least two vertices. Show that G contains at least two vertices of the same dgeree. ( ‘) L h\ 9 7; 4 V x... ” . V10 lea-Y5 DJ“ ega‘z'b M C “MCEN‘OQ‘ > l " deg Cor U 6 V he]? CL‘l' . 92> ?18zom\‘\s>\2_ ?l LWC{?\Q .‘ QMME h Value} £50m I]; . [ \E CLO]— ‘bwo musl— $2., [—le Same. [level—E. G Mob'l“ walracivt Oil— \ecx$lr ‘l'UJD uerldgzg 0k llng gqmq dcareg’ [3] (c) Let G: (K E) be a connected undirected graph. What is the largest possible value for IV] if = 30 and degfil) 2 5 for all 'u E V. We, loaves : i < 60 : 1.]E[ e Zdzabl) / > W,- 4‘ uéU be ' 0.5 ¥U6V2a€é(udab MA CM 201 Page 5’ of 4 3. Graph Theory I] [4] (a) Let G n (V. bemauloop—free connected graph with |V| 3 11. Show that either G or its complement G must be non—planar. Lch Exp/(é) Mm rages“ a {abuse ‘hia‘EeV‘Qakb. ' :(u a55— <=* % éed‘i‘a‘l i“ Kit (‘0) -U; G— {S planar‘ lEi 9-3-1“ “’6 r: '13- .(I; g is, Planar\ 1'é\ g 3-iV\—-6 :: 2} "" : +1} _—. 8‘14 ' if; huh“ G amd G are. Piavw‘r \ 'S I which ww’iai be. 0; Camifadici'im LAX-HA (by * Rafi“:- IJD‘EK Q and G Cdmnoi” bv. P\6W‘ar- [3.] (b) How many Hamilton paths does the graph Kan“, where n E N, n 2 1, have? Remember the paths v1; 112,123 and 123,112,15 are considered to be the same. - eawbx “Qwihhow path MUS? skewi— ‘m (We, Qatar i’fom Quad Mcmce', wiu'qigb (and in MS - ECLOR WWEA- Khhafhait utf‘in'cfilko 490M SPWO Ycu‘ RKO‘KQ I we; put-idle. h +\ WKAS ,u» n “MAS prode—h (‘11 6‘3 ' We. CCMA CAN-0.931. ._\\—— ’BVA mil.— n wqag d I ma, 3 (amen. 01ft. Comm zfe M\\'" Li __\1_., “‘4 a LAN: Scum/“L 0 M01 . kiAg ‘5 : “wagit gr L M ix (h+13‘.~h\. . ,_ 3 KM“) ‘o “ (WU-(M3 (w 1—) \ \ —. W MA CM 201 [3] Page 4 of 4 (c) Determine the chromatic polynomial of the parachute graph Y”, n. > 3. The parachute graph Kn is the graph obtained from the cycle C” by adding two new vertices joined by an edge and joining both by an edge to all vertices of C”. You may use the fact that P(Cn. A) : (A m1)”+(+1)”(/\ 71). Beiow is depicted Y5. fih coioirfi Um:- COUA color: cu Ubiél/x 7t “"433 la in (vi—i) wands. Wa— M01 (7“?) Rance. chie. cam FLCLMIAFZU was +9 CD tar IE C9106 5 (cu/xiii“; V13 Rance. x: 'h-UHW- 13(Cni’kdld who [ox—23" «~ M“ {WW} ,— ...
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