practice_test3_solns

practice_test3_solns - -' ' “sectien- ] it /_. ‘ (36)...

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Unformatted text preview: -' ' “sectien- ] it /_. ‘ (36) countereXample to demonstrate this. ( l V = number of vertices, | E | = ninnber of edges ) 1. Determine Whether,th'e"statement is true for all (simple) graphs, andif notpgive a ' r .- (i) If G is iseniorphictto a bipartite graph, then G. is also bipartite. '- .‘ Trtie False x (ii) IfIVl=7,and|E|=12,menGmustbepmfi:_- u True False (iii) If each vertex 9f G has degree at least 1[/21 V I , then G is connected. Fizeiez micjiilgor hear-K I. a r ' a}? vertex V q — , {fl {yutZ-‘S‘ I W ("fur , ‘ehé False _ / .' fix, €44.21 ‘ £32373? i’flw'f' I C atlwfif E3 fifi‘w. I \ ~ ~ I 9 “£3.6ij 5665" 3%.. £4} fixl ‘1'} V; I , vr ‘ Y‘UW‘JBQF @fi'fi’fflfi True False leflirii‘é? C" “1359.91 . W 3% £th {3 5pm m“ i- n c . ‘ lf ‘ (v) If each vertex of G has an even degree ; then [ V | is even. i * c/C t i r. w, ‘ l V l 3; bjmcf‘t.‘ .5 True False 4 w ; OCR 5 (vi) If G is connected, and its edges have weights, all of which are different, then G has exactly one minimum spanning tree. ' ' ' Kb?!“ Ky" LAS- iccsl ‘5 _ LI“ 1' i 3V? ) }.-'\3'“§,.._e‘{- ' True SB mama" 5 dun/5 (,le (Le; _ V 2.Fo‘r which po—sitive integers n does file complete-bipartite graph _ .. 613) _ ' ' - , (i) , have an Euler cycle? firm-(Sir 1w: “3 {mwa t: mew-K .. .j.L—-v--~-C‘-' (iii) pave a pluanarxdrawing ( i.e., is a planar graph). 4; .I I =/, _“ 1 _7 ‘I '. _ ,, ‘ _, D ‘3 kiwi-ex Jew-SAL» SJ Eiflvw 1: cvin G (i)- Find the chrOmatic number of 6. (ii) Find the‘chromatic number of H. (iii) Are any two of the three graphs shown isomofimic? EXPLAIN. =1 M 1 x , i ' - .1.“ §— “1 ,u ' Xx“); LHGUV {f if s ‘b‘ The ILIV‘w‘n{ \Wwwm A L 2:: .. r r -r-’ ‘3‘ v.12- 1 3 1’ r“; _ s 5 3 J} i {-u ., ‘~ 2}" L.="7 3.x \, Math 26502 131,132 Name- 1. Determine whether the statement is true for all (simple) connected graphs, and if not, give a counterexample to demonstrate this. (A(G) = max. degree,‘7C (G) = chromatic number, [VI = number of vertices, [El 2 number of edges) (36) . ‘ (i) If X'(G) _<_ 4 , then G is planar. 5%. K33 W 1: '3» - " __ L ' t 2 True False ii) IfA (G) = 4 ,‘ then 7((G) g 4. E . Kg @ a / R 1 g) A = q True False iii) lfGis planar, then XKG) _<_ 4. ._., u. 'THG “ Faun. QoLon— l MW . True False iv) IfA(G) =4 , then 32W] 24a: Elms .<. Wl-‘l True False v) le(G) 5 4,then£\(G) g 4. E -s °§< 2:9. , A = r: l/ I True False vi) If G is bipartite and A (G) g 4, then IVI _<_ 8. W / x ._ 1 h“ __ 0' True False __ .. I - .1, “Lia—6»: .; _ K». can _, .H .7 _ w ‘ r 3 3 ‘ 2r 1 I i) Exactly one of these graphs is bipartite. Which one is it? I 3 ii) Find :mA). - -7 __Li__. iii) Find 76 (B). ' _§_ iv) Exactly one pair among these five graphs are isomorphic. Which are they? Explain. A E ISGMWFHW “LM‘HS NINE THE- SAME QMMMA‘T‘C— ’ I NUMBBL’S’Qb 13 mm.- (Sawmilth Tn pow 0F mg 07146131 km: A Mn: E QA—N mm- Be LSQMOILPHJC. Pa Dumb E News Queue: 9F size Li, B mxmc som‘T: go :1- IS RMBC OK RN01: E". Cue-cc. mm A Mob 5 MU? :so MORPHAC. 3% thwcx RN RNMPMNFE Leawima. OIL ewe-“1K Thian XCCX=L11meJeTXCB\=3J 0R C £5. PLPx-Nfift! ULJHtLE B is met b.) A C/ {3'1 :3 Q. n: D H B s: B U} "U m D E in (IQ {If CD (D E. H D“ (D »8 :3" S '3" 0 Ln (1) (D D. Uq ('D S: (D H (IO :5" H (n 33 HQ H < ('D :3 0" (D .._. O S A U r-1 p: E 3:. v (16 _.A.. ..._ ..._. * 4 3 6 r 5 * 2 4 __ _ ..__‘_ "B" 4 * 2 7 ' 5 5 * =e< ‘ ‘ -C 3 7 2 * =1: 1 3 a: 6 D 6 7 * * l a: =z< 5 E r 5 5 1 1 I * 4 ;7 - a: F * 5 3 =£< 4 a: a: 2 G 2 * * * 7 '1: =1: 8 H 4 * 6 5 =2: ' 2r 8 >3 ‘ B (W me z: t‘—{\ (ii) Show that between any two vertices there is a path of total weight at most 8. Luca Air Tue- m! A SPfi-NNtNGa may Qnmmcms A LUMQUQ PATH muses!“ Adv! SAUL BF me VMTLLGS‘ In “nu-s enge- emw one 0F moss Perms HAS Tom vet-hem- MOM? WAN 8’. Tam Ls THE Pfl—T‘H FMM 6: mo H. Rm- 6. mm H Aire— TOthhB err .w shee- 0F mafiamT 3? (Mb Etc-1 me:- Ffim G_R._Ru-OF Tomk wane-Mr £03, ...
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practice_test3_solns - -' ' “sectien- ] it /_. ‘ (36)...

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