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Mohamed G. Gouda						     CS 311
Spring 2015							     Midterm 2
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1. (5 points)
What is the chromatic number of the complete bipartite graph K2,3. Explain 
your answer

2. (5 points)
Let G1 be a connected graph that has a vertex v1, and G2 be a connected 
graph that has a vertex v2. Also let G be the graph that is constructed from
G1 and G2 by connecting the two vertices v1 and v2 by an edge (v1, v2). Give
an argument that G has a path that connects any vertex u1 in G1 with any
vertex u2 in G2.

3. (5 points)
Let G be a graph with 5 vertices.

(a) Explain why the degrees of the vertices of G can't be 1, 1, 1, 2, 2
(b) Explain why the degrees of the vertices of G can't be 0, 1, 1, 1, 5

4. (5 points)
Consider a graph G = (V, E) where V={1, 2, 3, 4} and E={(1,2), (1,3), (1,4),
(2,3), (2,4), (3,4)}. How many edges are there in any subgraph G'= (V', E') 
of G where G' is a tree and V=V'. Explain your answer. 
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Solution:
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1.
The chromatic number of K2,3 is 2. An explanantion of this fact consists of
two points:

First, K2,3 is a bipartite graph and is 2-colorable (by the Bipartite Graph
Theorem) and so its chromatic number is at most 2.

Second, K2,3 has several edges and so its chromatic numnber is at least 2.
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2.
Because G1 is connected, then G1 and G have a path (u1, ..., v1). Because 
G2 is connected, then G2 and G have a path (v2, ..., u2). Therefore, G has 
the path (u1, ..., v1, v2, ..., u2) which connects u1 and u2.
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3.
	
(a) The sum of dertex degrees 1, 1, 1, 2, 2 is odd (7) and it should be
even by the Handshake corollary 1. Also the number of vertices of odd
degrees is odd (30 and it should be even by the Handshake Corollary 2.

(b) Every vertex degree in G should be at most (n-1) where n is the number
of vertices in G.
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4.
Because any G' is a tree, the number of edges in any G' is equal to (n-1),
where n is the number of vertices in G'. Moreover, because the number of
vertices in any G' is 4 (which is the same as the number of vertices in G),
The number of edges in any G' is 3.
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