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Mohamed G. Gouda						    CS 311
Summer 2014							    Homework 2
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1. (4 points)
Let G be a graph whose maximum degree is k. Assume that three vertices, whose 
degrees are less than k-3, and their incident edges are removed from G 
yielding a graph G'. 

a. What is the largest possible value of max-deg(G'). Expalin your answer.

b. What is the smallest possible value of max-deg(G'). Explain your answer.
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2. (3 points)
Complete the following three sentences:

a. One way to show that a graph G can be colored using at most k colors is to
show that ...

b. One way to show that a graph G can be colored using at least k colors is to
show that G has a subgraph G' such that G' has k vertices and ...

c. One way to show that the chromatic number of a graph G is k is to show that
G can be colored using k colors and ...
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3. (3 points)
Prove by direct inference the following predicate

	(G is a graph with n vertices, e edges, and k connected components
	 where each component is a tree) => (e = n-k))

Solutions:
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1.
a. The largest possible value of max-deg(G') is k.
b. The smallest possible value of max-deg(G') is k-3.
 
Explanation: Graph G has a vertex u whose degree is k. Because the degree of
each of the removed vertices is less than k-3, none of the removed vertices,
say vertices x, y, and z, is vertex u. 

Note that:

1. if graph G has no edge (u, x) then removing vertex x and its incident edges 
does not reduce the degree of vertex u.

2. if graph G has no edge (u, y) then removing vertex y and its incident edges 
does not reduce the degree of vertex u.

3. if graph G has no edge (u, z) then removing vertex z and its incident edges 
does not reduce the degree of vertex u.

4. if graph G has an edge (u, x) then removing vertex x and its incident edges 
reduces the degree of vertex u by one.

5. if graph G has an edge (u, y) then removing vertex y and its incident edges 
reduces the degree of vertex u by one.

6. if graph G has an edge (u, z) then removing vertex z and its incident edges 
reduces the degree of vertex u by one.

From 1, 2, and 3, we conclude that the largest possible value of max-deg(G') 
is k.

From 4, 5, and 6, we conclude that the smallest possible value of max-deg(G') 
is k-3.
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2.
a. G has k vertices

b. G' is a complete graph

c. G can be colored using at least k colors
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3. 

	(G is a graph with n vertices, e edges, and k connected components
	 where each component is a tree)
=> {let each component c.i have n.i vertices and e.edges}
	(n = (n.1 + n.2 + ... + n.k) and
	 e = (e.1 + e.2 + ... + e.k))
=> {each connected component c.i is a tree; thus e.i = (n.i-1)}
	e 	= sum(e.i)
		= sum(n.i - 1)
		= sum(n.i) - sum(1)
		= n - k
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