csrn Math D' Arcy McGee High School Dec 5,2016C2 Teacher: Mr. Randimbiarison 50Ut 7"O,Js

Graph TheoryMajor Assignment

84Duration: 60 minutes

NAJJE: _ SECTION: MCU504

1. Erica wants to change the graph on the rightinto a tree diagram. (4 marks)

Which two edges of the above graph must be removed y-roduce a tree diagram?

A) QPandRV @ QRandSTB) TU and RS D) UT and VR

2. Which one of the following graphs contains the longest simple circuit? (4 marks)

A) ? C) 0 p.-.-~ .---/ O_s- R -D. ~p-0.----

Q-R-f-5-c9 \"~~ // .•.<, »: \. lwfl- ~7.>:

L~jth= 4,/ ,..""'-. \~L.,..r._r "

R ./ s· R Q

B) p Q @ T R.•S- P- R-Q.- T-U··R.'-'"<, b(ilt=7-'......••..

f-G -R-s-f,.. '"'---•...•.."

'-.•.... kl\:;~z: 4-"-,...

R S R U

CSTll Math - Discrete Mathematics - Graph Theory - Dec 5th, 2016 Major Assignment Page 1

3. Euler path/circuit, Hamiltonian path/circuit. (12 marks)

Graph 1.

a)

(2 )

c) Does an~r circuit exist in the graphabove?~1 NO (circle yes or no)

....!!..':l...JL...!...!-.!.....1...-"'~~~..<I....!.-""--"""P--tl~~o Explain: 7ho€. deJjfefl cJ?al(U~ vu-fl'c ~QIe- €VQ..n •

___________ (2 marks)

If Yes:Starting vertex is: 3 OF 2Ending vertex is: ~ 3

(1 mark)(1 mark)

b) Does an Euler circuit exist? YES I@(circle yes or no)

•Explain: 1D ()I:,laj~ on Eul-er tirju/t-QJI VRrf/ces must be Of'..even dR!Jre e. .

___________ (2 marks)

If Yes: i J /11Starting vertex is: --MaEnding vertex is: . ,

(1 mark)(1 mark)

___________ (2 marks)

d) ~ a Hamiltonian path exist?~~.sI NO (circle yes or no) (2 marks)-...- .

If YES, write that path beJow; rf K-l-hJ-Y1-i9-:,-h-j-,_-e.-d- L_ b a.-jl- -

The length of that path is: i S-

CSTll Math - Discrete Mathematics - Graph Theory - Dec 5th, 2016 Major Assignment Page 2

8(3)

'~·P(4) \

\ b) Which two edges must be removed to obtain an Euler path?.H(3). \ Cl',)

'.;1/'/ l4;The two edges to be removed are: and /2

/""-~-,..,..,

4. Given the graph below (10 marks)

fl urn ,'"(+0 n I ane. re LLL:t:

c4) A .4"'-----"'--"'::

G'H £G-F

a) Identify a Hamiltonian circuitTrace that circuit over the graph and write it down over theline below

A- B- G-I-l- C-]) - £ - f -~ 12

.f(-

c) Where will be the starting and the ending vertices of thatEuler path after removing the two edges above?

The starting vertex is: _ 11 ¥-11 .:xThe ending vertex is: _

d) Determine the distance d ( A, C).

dCA, C) = 2- ,~{8Iwrf1.orrfi'isA-p - c.. er/l-!3-C

e) How many edges must be added to the original graph inorder to obtain a complete graph?

..., "fhe£"t are g ifR.,rf-l'ces , ~ ~""0~ The AiU'Yl r1 ~ e/A I~ Gt. Corvf e. r-rtt~

.~. g v~rH' C£A 1.5 8~7 =- Sf:,>Th-e lUuYl b.u cf ~d~ I s S (, z: 2.8 12

wrre,yj-~ j~U~ Qfe fLf ~d~

T~{e,ILj more eci8"smusC L cvtJedi'n".le. to ,,/ofart1<A OJmpMi r;r~.

12

CSTll Math - Discrete Mathematics - Graph Theory - Dec 5th; 2016 Major Assignment Page 3

5. Write a conjecture about the relation between the chromatic number and the number of vertices ofa complete graph. (10 marks)

Use at least three examples to support your conjecture:

1st Example (2 marks)

.:(f v..trt-(' (00 '::: 2d1lc11h~; L~.:;::. z.

2nd Example (2 marks) 3rd Example (2 marks)

#- iftrit'CeA :::::-3dri,'IJ)--eJe c. 1-.lj,{YI-~-r z: "3

tt /if'fr cu: 4~\or...~\c it:= 4.

The ~onjecture is: .In lA CtJ.n.-# ~ I 1h n/;tf'l beT 01 V..er f t' ( ~

15 .ortJ 1o·1J.p cA ru-rnatic nu rY1 her.

__________________________ (2 marks)

Based on the conjecture found above, determine the chromatic number of a complete graph thatcontains 6, 834, 287 vertices.

(2 marks)

CST11 Math - Discrete Mathematics - Graph Theory - Dec 5th, 2016 Major Assignment Page 4

6. A graph is defmed by the relation R, such that "x +y is a multiple of 3", where x and y are

Vertices, which are defmed by the set V= {1, 2, 4, 7, 8, 11}. (14 marks)

a) Draw the graph that illustrates the relation R. (4 marks)

tOLb~r5

b) Determine the chromatic number of the graph found in a)

The chromatic number is: -=-2_· __ (2 marks)

c) Identify the longest simple circuit and evaluate its length

The circuit is: _2_-_1-,--_11_-_4_' -_8_-_7_-_2- (2 marks)

The length of the circuit is: _~b~(2 marks)

d) Show if an Euler path exist. (4 marks)

In orc).tr·f o e ~/N~ 0-11 E« lev- rf?:JJ,~ fh.er~ masl-~J? Q)(aci7 __'two .v-€rf/c~ w;1A odd ~~e(;L ,LJh/c.h /5 no I the. case lur'e.-(all ~ tkm are. {),{caJJ 'cb1r~CL)

CSTll Math - Discrete Mathematics - Graph Theory - Dec 5th, 2016 Major Assignment Page 5

7. Consider the graph below: (12 marks)

b) Draw a tree of maximum value of the graphon the left and determine its total weightvalue. (4 marks)

Drawing:

F

a) Draw a tree of minimum value of thegraph above and determine its totalweight value. (4 marks)

Drawing:

A'f

B

rG- t

30+3r+4tJ +4':>+ sv-fSiJ -=:.02~)Total weight value of the tree is: ~Jj.s--

c .50\71>

G-

IOO+qO+gDt){)+~S:-- 4/)- ./

Total 'weight value of the tree is: ~

t

c) Determine the chromatic number of theoriginal graph seen above on the left.(4 marks)

Show your work:Cl.)

A~ __ ~

\0 ~(~)~~ :/ \.0J An 35 J:!Io Cl.·)

~ 50(~ ~-------+·I

3The chromatic number is: ---

CST11 Math - Discrete Mathematics - Graph Theory - Dec 5th, 2016 Major Assignment Page 6

8. Determine the path of minimum distance from A to G (10 marks)

The weightings on the edges are distances in kilometres.

Show your work

f'Jt 1.: A- c - L - J< - F -- G14·-t \7+ c}+I,>+& - 58

Po1h :l: A _ E.- L - F - G-IS-i 11~ 10-+ g ..- 44-

feat 3:A-s-F-G-IS-:t- 2, + '8 .- 46ratL. 3 A _ t: - L - F- C-14·-/- /'2+ 10+ 8 := 44

The path of minimum value is: A - E - L - F - ~The total distance in kilometres is: 4'1= km5

e-r A -c- L - F-G-

CST11 Math - Discrete Mathematics - Graph Theory - Dec 5th, 2016 Major Assignment Page 7

9. The steps to complete a series of tasks of a given project are represented by the diagram below.The tasks are represented by the letters. The weightings on the edges represent the time in minutesneeded to carry out a task. (8 marks)

C

A~H.~. 1"\.. 12 ·~6

.~~. F 3 .

8~-~ .••• ~

./C'~~B

a) Determine the critical path of this project (4 marks) . ~ ,..., JFtJj' : slli r f _ /1- C - H - I - b,wL, fti/J..? t sfitl- B_c - C--1-~-:-

.g.-( ~ -+ 6 t 4::: 27m-t-fI to f er + 1i: 4 z: :52171/"p&J1 !) !YlKI -It -p.- F - I - e-:« .

~ tl2.-·~ '3 T 4 zz: 2. 7t?J1 t1

PdL J ; vlruC-A- E.- F - 1- - 0n-J.-.. '1.' ,~ -t q + .> -r '-( := oii;11o 1t1 ,#

fill '-i' 'frfacl-A - e- (;- r - ~d- .~, ~+ Cf -+ q·f 1-; )Dt1ll(1

ftJts! ~f-B~E-F-r-~ ,10 + q + .~+ t:j := Z{,rnln

fall. G ; ~/wl~ f3 _ G - J>- b.-cL10+ er -1- 4 = 23 rru~

b) Determine the minimum time needed to complete the entire project. (2 marks) "

'~W'I1I~ ft;1,e ~;:d !o ~ -a -h1~'nL)r,/cf•

J S 3;< MIr1

c) Because of a new constraint, the time needed to complete step B is reduced to 7 m/a • Doesthis new constraint affect the time to complete the whole project? Justify your answer.(2 marks) ,

ShrlJ ~ Iv /171.:" -7 /?dLs -;:. ~ud /" ~{-3:= ~3rw:;PttP G -? ('~td-lf) ~3 -~;:= &_D ~

_' ,• . f¥i 7 - n.d<u<.i. to iP·- 3 ~ :J-'1 .. --7' (//U ~IM (tI: .Cnt:~ .p;..#- ,~?.ft.C-.t-R.et .: r. \ M;..'\

~~ rr.tW cc~I" 'cJl fp::/fi "5 ~fclfr/•. tJ _ e -0- -J _ ~cL -? ~0 '{YIM1( tr''''''I

CST11 Math - Discrete Mathematics - Graph Theory - Dec 5th, 2016 Major Assignment Page 8