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QUESTION 9 Nd, 2 d2 2 d3 2 ... 2 d. 21 be a sequence of positive integers. Consider the statements below: St 1: There exists a tree Twith n 22 vertices such that the degrees of the vertices are du, den St 2: dj + d2 + ... + dp = 2(n-1). VEV We want to prove that St 1 - St 2. Note that we can prove St 1 + St 2 by using the theorem about the sum of the degrees of a graph T=(V.E): deg() - 2 LEI (You are NOT required to prove St 1 - St 2.) To prove the other implication st 1 st 2, we will apply induction on the number of vertices n according to the following steps. (a) Check the basis step for n = 2. (b) Write down the induction hypothesis for some k 22. (c) Let n 2 3. Show that if de + d2 + .+ dn = 2(n-1) and d/21 for each I, then d - 1 and dy>1. (d) Apply the induction hypothesis to d, -1, dz., dk and deduce the desired result. Complete the steps a,b,c,d to prove that st 1 st 2. TT T Arial 3(12pt) 3's QUESTION 9 Nd, 2 d2 2 d3 2 ... 2 d. 21 be a sequence of positive integers. Consider the statements below: St 1: There exists a tree Twith n 22 vertices such that the degrees of the vertices are du, den St 2: dj + d2 + ... + dp = 2(n-1). VEV We want to prove that St 1 - St 2. Note that we can prove St 1 + St 2 by using the theorem about the sum of the degrees of a graph T=(V.E): deg() - 2 LEI (You are NOT required to prove St 1 - St 2.) To prove the other implication st 1 st 2, we will apply induction on the number of vertices n according to the following steps. (a) Check the basis step for n = 2. (b) Write down the induction hypothesis for some k 22. (c) Let n 2 3. Show that if de + d2 + .+ dn = 2(n-1) and d/21 for each I, then d - 1 and dy>1. (d) Apply the induction hypothesis to d, -1, dz., dk and deduce the desired result. Complete the steps a,b,c,d to prove that st 1 st 2. TT T Arial 3(12pt) 3's