pset5 - EECS 310, Fall 2011 Instructor: Nicole Immorlica...

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Unformatted text preview: EECS 310, Fall 2011 Instructor: Nicole Immorlica Problem Set #5 Due: October 27, 2011 1. (25 points) Let ( s 1 ,s 2 ,...,s n ) be an arbitrarily distributed sequence of the numbers 1 , 2 ,...,n- 1 ,n (i.e., a permutation). For instance, for n = 5, one arbitrary sequence could be (5 , 3 , 4 , 2 , 1). Define the graph G = ( V,E ) as follows: 1. V = { v 1 ,v 2 ,...,v n } 2. { v i ,v j } E if either: j = i + 1, for 1 i n- 1 i = s k , and j = s k +1 for 1 k n- 1 (a) Prove that this graph is 4-colorable for any ( s 1 ,s 2 ,...,s n ). 1 (b) Suppose ( s 1 ,s 2 ,...,s n ) = (1 ,a 1 , 3 ,a 2 , 5 ,a 3 ,... ) where a 1 ,a 2 .... is an arbitrary dis- tributed sequence of the even numbers in 1 ,...,n . Prove that the resulting graph is 2-colorable. 2. (25 points) An open Eulerian walk in a graph is a walk that traverses each edge exactly once and returns to a different vertex than it started from. Prove that a connected graph has an open Eulerian walk if and only if it has exactly two vertices of odd degree.has an open Eulerian walk if and only if it has exactly two vertices of odd degree....
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