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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.
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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
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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,
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[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... ”
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Cor U 6 V he]? CL‘l' . 92> ?18zom\‘\s>\2_ ?l LWC{?\Q .‘ QMME h Value} £50m I]; . [
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[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.
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' 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
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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:
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- Spring '09
- MarniMishna
- Graph Theory, Planar graph, Glossary of graph theory, undirected graph, MA CM