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Quiz5_Solution

# Quiz5_Solution - MATH 2602 ~ Quiz 5 — Name Each problem...

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Unformatted text preview: MATH 2602 ~ Quiz # 5 — Jone 17, 2009 Name: Each problem is worth 10 points. Show all of your work, and try to be as neat as possible. Points may be deducted if solutions are unclear. Use the back of the paper if you need more space. Problem 1. Prove that euery simple gmph G with n uertices contains at least two uerttces with the same degree. (Hint 1: Use pigeonhole pn'nez‘ple. Hint 2: This problem should seem familiar!) _ ‘ _ ‘ : H (“K a a PC6047. bus Lndodﬂon’ @5068 (Doe \Kk‘?) €\'*\\r\e< w “\ML g‘kﬁe-me‘fﬂ “GKOPKS. . mcxuwxse 1:39; Asswe 5mm. ~90; Mn. _ LQ’V WWW”; W - \r‘? GUS connecjvtd‘k‘nen th‘ce are \M)‘ qetr'ﬁces 0% Ae (96 b 'e‘b ‘thVSBE, 68.9fm)? i1}...Je\~.t§ w'o’h '0 vecﬁceg \ WiC‘neKGS {or acareesmw ‘ ‘ (39th Q S Woe MUS'Jr'be A030 0‘: “Ha some ééf‘fe?‘ thﬁﬂi : ; mm (Wmseu stake 0\ (onOercd (ﬂowgone-M (3‘; Cut ea] \egg We ﬂu e}? WEH'WHS com onenJr most Mfg; «JAIL stemmed? Name 3t: 6993 (-51 ' Problem 2. Cons er the ﬂoor plan of a. museum low. Each letter represents a room. M U \$60M The marking on the walls are the doors from room to mom. . (a) 3"311’7’9-‘567!t the ﬁOOf‘ plan as a graph- What is your vertex set? What is your edge set? (I?) Is your graph bipartite? If so, what is the desired partition (that is, what me V1 and Vg)? If not, why? (:2) Imagine the entrance and ens-it to the museum are through room A. Is it possible to enter the museum, walk through each domey exactly once, and then emit the museum? If 30, show how you. would accomplish this. If not, give a mathematical reason. "I mg _.._....C . \Kbﬁf iA(\$/610;E{ Ebﬁulg ‘1‘ __ FL (its): egs‘e'lﬁfsﬁﬁ{A‘D'SﬁEFEECE'E . C(l: b t I; ,,,,,,, ‘\- Q0151 ﬁt?! (“‘3 "L E‘ Hgﬁug’éﬁﬁjﬂ f5 ,_ Them 0&2 "ka 090% GONE;- CL3¢\P.S {h <\’\'€ C3 \r\.\$ WAHCMKS A‘CIEFWQ "traumas-"g b) U368. \yqéé‘QHC \M’é 6W 93‘ so “the answer-<3 No. MATH 2602 — Quiz # 5 — June 17, 2009 Name: Each problem is worth 10 points. Show all of your work, and try to be as neat as possible. Points may be deducted if solutions are unclear. Use the back of the paper if you need more Space. Problem 1. Prove that every simple graph G with n oertices contains at least two eerttees with the some degree. (Hint 1: Use pigeonhole principle. Hint 2: This problem shouid seem familiar!) ASSUW‘E ﬂwrre QMKFK \llweVUDj') “3) dffilv):bl do (w): h—i w View Luis ‘3“‘W\;'<"5> m “3‘ ‘ (“003‘s md‘meme ‘L‘fb @ﬁ‘gbﬂf‘kﬁx) WﬂkV%\\C‘3§\bvy‘:‘D Ag my: Hoax/1'33 fox Q3 \iC-VCQ) m“ \$9 MDE‘CC’r‘f‘Qi 1%? e,.Me(\j \IEVCKQ. EHMQ( was AHWQ We ”0. next—“Q95 (es wkkf “Q (Q sci-We ‘R'AI’P omo\ mi emailed 3*" . «\\Q (a ewe Amo' eeﬂjh‘ces A WW? Seemedeﬁreﬁ Problem 2. Consider the ﬂoor plan of a museum below. Each fetter represents a room. M U SﬁU M The marking on the walls are the doors from mom to room. _ (a) Represent the floor plan as a graph. What is your vertex set? What is your edge set? (5) Is your graph bipartite“? If so, what is the desired partition {that is, what are Vl and Vg)? If not, why? (6) Imagine the entrance and can't to the museum are through mam A. Is it possible to enter the museum, walk through each domey exactly once, and then exit the museum? If so, Show how you 2:10qu accomplish this. If not, give a mathematical reason. ...
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Quiz5_Solution - MATH 2602 ~ Quiz 5 — Name Each problem...

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