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COMP2121A: Discrete Mathematics
创建日期：2024-05-17 15:54:24
所属分类：数学mathematics
标签： COMP2121A
Discrete Mathematics

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COMP2121A: Discrete Mathematics

Rules: Discussion of the problems is permitted, but writing the assignment together is not
(i.e. you are not allowed to see the actual pages of another student).
Course Outcomes
• [O1. Abstract Concepts]
• [O2. Proof Techniques]
• [O3. Basic Analysis Techniques]
1. (10 points) [O3] A path of a graph G = (V,E) is a sequence of distinct vertices
(v1, v2, ...vk) such that (vi, vi+1) ∈ E for i = 1, 2, · · · , k − 1. A longest path is a path
P such that no other path has more vertices than P . In this problem, we suppose G
is a connected graph with |V | ≥ 3.
(a) If G has 2 longest paths, will they share at least one common vertex? Does there
exist some s > 2 such that s longest paths in a connected graph do not share
a single common vertex? (Hint: you can consider the following graph, trying to
prove that all of the longest paths of this graph do not share a common vertex.)
(b) If each vertex v ∈ V has degree at least |V |2 , what is the minimum possible length
of the longest path of G?
2. (10 points) [O3] In a graph G = (V,E), a matching M in G is a set of pairwise
non-adjacent edges, no two edges share common vertices. A vertex cover V ′ of G is
a subset of V such that (u, v) ∈ E ⇒ u ∈ V ′ ∨ v ∈ V ′, that is to say, it is a set of
vertices V ′ where every edge has at least one endpoint in the vertex cover V ′. We
denote α the maximum size of the matching of G while β the minimum size of the
vertex cover.
(a) Prove that for any G, α ≤ β.
(b) Find an example that the equality holds in 2a.
(c) Prove that for any G, β ≤ 2α.
(d) Find an example that the equality holds in 2c.
(e) Find all situations that the equality holds in 2c (Hint: you may use the following
advanced theorem (Tutte-Berge’s Formula): If k is the minimum number of
1
vertices not covered by a matching, then there exists a subset U ⊆ V such that
the induced subgraph on G − U has |U | + k connected components with odd
vertices.)
3. (10 points) [O3] Let G be a k-regular graph with n vertices.
(a) If k = 3, what values can n take? For any fixed positive integer k, what values
can n take?
(b) If k = 3, will G always be Hamiltonian if n = 6? Will G always be Hamiltonian
if n = 8? Please explain your reasons or give a counterexample.
(c) A perfect matching M of a graph G = (V,E) is a matching that covers all
vertices, i.e., a subset of E such that each vertex v ∈ V appears exactly once as
an endpoint of one of the edges in M . If k = 3 and G is Hamiltonian, show that
E can be partitioned into a union of three perfect matchings.
(d) Let k = 3 and G be Hamiltonian. Assume every edge in G under this case lies
on an even number of Hamilton cycles. Using this fact, deduce that if k = 3 and
G is Hamiltonian, then G has at least three different Hamilton cycles. In this
question, a cycle is identified solely by the collection of edges it contains; there
is no particular orientation or starting point associated with a cycle.
4. (10 points) [O3] A graph G has n vertices. For every 4 vertices in G, there is a vertex
serving as the neighbor of the remaining 3 vertices. At least how many vertices are
there neighboring all other vertices in G (i.e., vertices with degree n − 1)? Please
explain your answers.
5. (10 points) [O3] Let m and n be two positive integers. We construct a graph Gm,n
as follows:
• draw m disjoint copies of K2 , each containing a red vertex and a blue vertex;
• add n yellow vertices and n green vertices;
• join each red vertex to every yellow vertex by an edge;
• join each blue vertex to every green vertex by an edge.
(a) What is the chromatic number of Gm,n?
(b) Find all m and n for which Gm,n is Eulerian.
(c) Determine whether each of G4,5, G6,6 and G7,4 is Hamiltonian.
6. (10 points) [O3] Let G be a connected planar graph with 120 edges. We can partition
the edges into two sets, S1 and S2 with |S1| = 90 and |S2| = 30. Suppose that,
∀e ∈ S1, the face on one side of e has 3 edges; and the face on the other side has 10
edges. Also, suppose that, ∀e ∈ S2, the two faces on each side of e are distinct from
each other and both have 10 edges. How many vertices does G have?
2
7. (10 points) [O3] Amy has a lamp in the bedroom, which shows n different colors and
the color of the lamp is decided by the states of 2 switches. Each switch has n different
states. The color of the lamp would change when both of the two switches change
their states. However, how the color of the lamp changes when one switch changes
its state while the other does not is temporarily unknown. Amy conjectures that the
color of the lamp is actually only dependent on exactly one of the switches. Please
prove or disprove the conjecture.
8. (10 points) [O3] The graph of the cube C3, sometimes called the 3-cube, can be
described as the graph whose vertices are the binary strings 000, 001, 010, 011, 100,
101, 110, 111 where two vertices are adjacent if and only if they differ at exactly one
digit. For instance 010 and 110 are adjacent because they differ only in the left-hand
digit. The vertices 010 and 100 are not adjacent because they differ at both the
left-hand digit and the middle digit.
(a) Does C3 have an Euler circuit? Is C3 a planar graph?
(b) Let C4 be the graph described similarly but with binary strings of length 4.
Thus, for instance, the vertices 0110 and 1110 are adjacent because they differ
at only one digit, but 0110 and 1100 are not adjacent because they differ at 2
digits. Does C4 has an Euler circuit? Is C4 a planar graph?
(c) Let C8 be the graph described as above, but now with binary strings of length 8
(so that there are now 256 vertices and each vertex has 8 edges). Does C8 have
an Euler circuit? Is C8 a planar graph?
9. (20 points) [O1,O3] Suppose Km,n is a complete bipartite graph whose vertices on
the left are indexed by {l1, l2, l3, ..., lm} and those on the right by {r1, r2, r3, ..., rn},
where m ≥ n ≥ 3. In this question, a cycle is identified solely by the collection of
edges it contains; there is no particular orientation or starting point associated with
a cycle.
(a) Let the size of a graph be the number of vertices in the graph. What is the max-
imum possible size of the induced subgraphs of Km,n that are Hamiltonian? Let
S be the set of all induced subgraphs with maximum size that are Hamiltonian,
what is the cardinality of S? Give your answer in terms of m and n.
(b) What is the maximum number of Hamiltonian cycles contained in any element
in S? Give your answer in terms of m and n.
(c) Let m = 10, n = 10. How many Hamiltonian cycles in Km,n contain the edge
{l1, r1}? How many Hamiltonian cycles in Km,n contain both the edges {l1, r1}
and {r1, l2}? How many Hamiltonian cycles in Km,n contain both the edges
{l1, r1} and {l2, r2}?
(d) Let m = n. How many Hamiltonian cycles in Km,n do not contain any edge
from {l1, r1}, {r1, l2} and {l2, r2}? Give your answer in terms of m and n.
(e) Let m = n. Suppose that M is a set of k ≤ n2 edges in Km,n with the property
that no two edges in M share a vertex. How many Hamiltonian cycles in Km,n
contain all the edges in M? Give your answer in terms of n and k.
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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