Mohamed G. Gouda						     CS 313K
Fall 2012							     Homework 2

1. Let G be a graph with 
       5 vertices of degree 3 each,
       4 vertices of degree 2 each,
       3 vertices of degree 1 each,
       2 vertices of degree 4 each, and 
       x vertices of degree 6 each.
Compute x if G has 35 edges.

Sol:
From Handshake Theorem,
70 = 5*3 + 4*2 + 3*1 + 2*4 + x*6
   = 15 + 8 + 3 + 8 + 6x
   = 34 + 6x

Thus, 
x  = 36/6 = 6

2. Let G be a graph with n vertices, where n >= 2. Show that 
(a) if G has a vertex of degree n-1 
    then G has no vertex of degree 0
(b) if G has a vertex of degree 0
    then G has no vertex of degree n-1 

Sol:
(a) Assume that G has a vertex v of degree n-1. Thus, vertex v is connected 
by an edge with every other vertex in G. The degree of every vertex in G is 
at least 1. Therefore, G has no vertex of degree 0.

(b) Because (b) is contrapositive of (a), then (b) follows from (a).

3. Generate all subgraphs of graph G=(V,E) where 
V={1,2,3} and E={(1,2),(2,3)}
Indicate which of the generated subgraphs is "induced" for G.

Sol:
There are 12 generated subgraphs; they are

G: the original graph		- induced for G

{1,2,3}, {(1,2)}
{1,2,3}, {(2,3)}
{1,2,3}, {}

{1,2}, {(1,2)}			- induced for G
{1,2}, {}
{2,3}, {(2,3)}			- induced for G
{2,3}, {}
{1,3}, {}			- induced for G

{1}, {}				- induced for G
{2}, {}				- induced for G
{3}, {}				- induced for G

4. Let G=(V,E) be a graph where 
V={1,2,3,4}, E={(1,2),(1,3),(2,3),(2,4),(3,4)}
(a) What is the chromatic number k for G? Explain.
(b) Define a valid k-coloring for G.

Sol:
(a) The chromatic number k = 3.
    The vertices 1, 2, and 3 need to have distinct colors, but
    vertex 4 can have the same color of vertex 1.

(b) color(1) = color(4) = blue
    color(2) = red
    color(3) = white

5. Let Kn denote the complete graph with n vertices.
Let Ln denote Kn after removing one edge from it.
(a) What is the chromatic number for Kn? Explain.
(b) What is the chromatic number for Ln? Explain.

Sol:
(a) The chromatic number for Kn = n.
    Every two vertices are connected by an edge in Kn. Thus, every vertex in Kn 
    needs to have distinct colors.

(b) The chromatic number for Ln = n-1.
    Let (u,v) be the edge that is removed from Kn to get Ln. Thus, vertex u can 
    have the same color as that of vertex v.

6. Let G=(V,E) be a graph where
V={1,2,3,4,5}, E={(1,2),(1,3),(2,3),(3,4),(3,5),(4,5)}
(a) Identify all simple paths from 1 to 5.
(b) Identify all simple circuits from 1 to 1.
(c) Identify all cycles from 1 to 1.

Sol:
(a) Simple paths from 1 to 5:
    (1,3,5), (1,2,3,5), (1,3,4,5), (1,2,3,4,5)

(b) Simple circuits from 1 to 1:
    (1,2,3,1), (1,3,2,1), 
    (1,2,3,4,5,3,1), (1,3,4,5,3,2,1), 
    (1,2,3,5,4,3,1), (1,3,5,4,3,2,1)

(c) Cycles from 1 to 1:
    (1,2,3,1), (1,3,2,1)