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Mohamed G. Gouda						    CS 311
Spring 2015							    Homework 2
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1. (2.5 points)
Let G be the graph (V, E) where
	V is the set {1, 2, ..., 5} and
	E is the set {(1,2), (1,4), (2,3), (2,4), (2,5), (4,5)}
Show that the chromatic number of graph G is three.
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2. (2.5 points)
Consider a graph G that has 11 vertices whose degrees are 2, 2, 2, 3, 3, 4, 4,
4, 4, x, y. Assume that the values of x and y are taken from the set {0, 1}. What
are the values of x and y? Explain your answer.
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3. (2.5 points)
Consider an induction proof of the quantified predicate:
	(All n, n >= 1, P(n))
where P(n) is the predicate (Any graph G with n vertices can be colored using
(max-deg(G) + 1) colors. Argue that the following "induction step" of this
proof is incorrect:

	P(n)
=> {Let G be any graph with (n+1) vertices. Because P(n) holds for G, we 
   conclude that G can be colored using (max-deg(G) + 1) colors. Thus, P(n+1)
   hold for G}
	P(n+1)
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4. (2.5 points)
Prove by direct inference the predicate
	(TR is a tree) => (All cycles in TR are of even length)
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Solutions:
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1. 
First, show that G can be colored using three colors:
	Color vertex 1 White
	Color vertex 2 Yellow
	Color vertex 3 White
	Color vertex 4 Red
	Color vertex 5 Yellow

Second, prove that G can't be colored using two colors: The induced subgraph
involving the three vertices 1, 2, and 4 is a complete graph and so it needs
to be colored using three colors.
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2. 
From the Hnadshake Corollary 2, the number of vertices whose degrees are odd
is even. Therefore, the values of x and y are
	either both 0	(in which case the number of vertices whose degrees
			are odd is two)
	or both 1	(in which case the number of vertices whose degrees
			are odd is four) 

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3.
The hint of the "induction step" is incorrect a s follows. Predicate P(n) holds
for any graph with n vertices. Therefore, P(n) does not hold for graph G, as 
claimed by the hint, which has (n+1) vertices.

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4.
	(TR is a tree)
=> {definition of a tree}
	(All cycles in TR are of length less than 3}
=> {length of a cycle in a nonnegative integer}
	(All cycles in TR are of lenght 0, 1, or 2)
=> {All cycles of length 0 or 2 are of even length and
    Exist no cycle of length 1}
	(All cycles in TR are of even length)
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