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HW MATH 192 components (let the sizes be k and n-k, where 12k 4 n-J 1) Chapter fl, Exercise 15 Maximum number of edges tr Lot W be a closed walk thet a_dis connected grap he - contains both x and y. 4 GH. ~ KCk-1) | (n—kVin-k-1) V, x 2 — - = zy ~ kK -k + M-kyY -M-k) W looks ike “ : 2 2 2 Nobo : somewhere u this wall , vertex = Rick tpt lok te cate y (5 encountered. n> —-n - Ink +2k~ at oo — Now, remove every time the walk repeats otae edge , Of remove a bop|= Bae If subwalk passes an edge twice The term kin-k) tS maximised Therefore, this Process always welds lwhen k= | f= | a closed troul / (4) + (*r | = (n- MAPO|=2} Beccutse detours thet begir and end ak the same vertex wrthout passing < Sa + I through 2 and y are deleted, both x and Y SULY inv the simplified This uso contradiction | ond Walk hence such CL graph must he connected 2) Choystes Il, Exercise 20 3) Chapter I, Exerase +t Suppose thet a oh of order with ot loast orion?) 44 (5 IA complete brpartite graph Kann clisconre cted has OL Hamilton cycle if WH IL | and m,n 2 2 Then, & must have ot least 2 A graph Ky q bas a Hamilton 6) Chapter I2, Exercise © path iff m-n| < | IF a graph needs k colors, +) Chapter I1, Exercise 62 there must be somewhere a set_of kK vertices where every let v be the vertex with degree p.| pour 1s adyacert | So, rt has p Neighbours Uy Uap, Up The smallest graph that needs Because the graph 13 a tree, there |k colors ts the complete qraph. are no cycles, 30 from each neighbor kK, , which has exactly (5) u,, there (s a branch qoung AWAY edges, from v that can never meet ay other branch. There ere, om graph with chromatic edges has at lLoast From v, p branches grow out ,and| $) edges each bruch must eventually end aloof ,S0 the tre has at least p}F) Chapter I2, Exercise 2 loaves. a) A graph with at lest 5) Chapter 12, Exercise + vertex cont be colored with 0 colors , $0 p, Co) = 0. IF we try J - colors, stcuting vertex colored red, the next blue, | Thus, constuck tenn of p. ck) and so-on , (because the cycle has }is 0. odd length) when you return to starting vertex, you cre fortec fo |b) If G has components 6, Gs, color a vertex the same cobras then. p, ck) =p, ck) p,. tk)... p__Ua) its neighbour ; 80 not allowed Each factor p, dk) has a lowest Therefore , 3 cobs wowd work, as |power k, thus the lowest term we could use a third clr for the|is kK’, 50, the coefficient. of RI last vertex. #0 sf c=! Cconnected) ~ XCC,) #3 =O if rot (disconnected) D Chapttr I), Exercise 26 because even y vertex has degree k, the sum of degrees is: nk The Sc of degrees (5 also =m, mM 1s the number of edges. nle=Im | m = DE As 6 ts planar, rt SKLIST LCS Euler’s (neq ucctlioty Me 3n-6 , N23 QS eo 3-8 k « 6- = Since W234 = 0,50 ki< 6 
