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184a_05w_2e

# 184a_05w_2e - Math 184A Second Exam(50 points 2 March 2005...

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Math 184A Second Exam (50 points) 2 March 2005 Please put your name and ID number on your blue book. The exam is CLOSED BOOK except one 2-sided page of notes. Calculators are NOT allowed. You must show your work to receive credit. 1. (10 pts.) Let G be the simple graph with vertex set { 0 , 1 , 2 , 3 , 4 , 5 } and edges braceleftbig { 0 , 1 } , { 0 , 2 } , { 0 , 3 } , { 0 , 4 } , { 0 , 5 } , { 1 , 2 } , { 3 , 4 } bracerightbig . Sketch G and compute P G ( x ), the chromatic polynomial of G . 2. (12 pts.) The local description of a decision tree for constructing sequences of A’s and B’s is given below. The notation BA S ( n - 2) means place BA in front of each sequence produced by S ( n - 2). S (1) A B S (2) A S (1) BA S ( n ) ( n 3) A S ( n - 1) BA S ( n - 2) Let S * ( n ) denote the entire decision tree. Thus S * (1) = S (1) and S * (2) has the three leaves AA, AB, and BA. (a) Prove that each leaf of S * ( n ) is an n -long sequence of A’s and B’s. (b) Prove that each non-empty sequence of A’s and B’s will be the leaf of some S * ( n ) if and only if the sequence does not contain two B’s in a row.
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