1. Let V = {cities of Metro Manila} and E = {(x; y) | x and y areadjacent cities in Metro
Manila}.
(a) Draw the graph G defined by G = (V; E). You may use initialsto name a vertex
representing a city.
(b) Apply the Four-Color Theorem to determine the chromaticnumber of the vertex
coloring for G.
2. Apply Euler’s Theorems and Fleury’s Algorithm to determine Eulerpath and Euler circuits in
each graph.
Euler’s Theorem 1:
• Every vertex of (a) has even degree. Therefore
it has at least one Euler circuit.
Euler’s Theorem 2:
• The graph has no odd degree, therefore, it has
at least one degree.
a.
Euler’s Theorem 3:
No of vertices
No. of vertices
w/ odd degrees
with even degrees
0
9
Sum
28
No. of
Edges
14
Fleury’s Algorithm: Path A-B-G-H-C-J-E-H-B-C-D-E-F-G-A is a Eulercircuit.
Euler’s Theorem 1:
• Two vertices of graph (b) haveodd degree.
Therefore it has no Eulercircuit.
Euler’s Theorem 2:
• Vertices with odd degree is >
2. Therefore it
has a Euler path
• The graph is connected and has just two vertices of odd degree,
therefore it has at least
one Euler path
b.
Euler’s Theorem 3:
No of vertices
No. of vertices
w/ odd degrees
with even degrees
2
10
Sum
42
No. of
Edges
21
Fleury’s Algorithm: The path E-F-G-H-L-G-C-H-M-L-K-F-B-EK-J-E-A-B-C-D-H is a Euler circuit.
3. A businessman has to visit five cities A, B, C, D and E. The distance (in hundred miles) between the
five cities as follows: If thebusinessman starts from city A and has to come back to his starting point,
which route should he select so that the total distance travelled is minimum.
Hamilton Circuits
Sum of Weights
A-B-C-D-E-A
7+8+9+8+4 = 36
A-B-D-C-E-A
7+5+9+7+4 = 32
A-B-C-E-D-A
7+8+7+8+8 = 38
A-D-C-B-E-A
8+9+8+6+4 = 35
A-B-E-D-C-A
7+6+7+9+6 = 35
A-C-B-D-E-A
6+8+5+8+4 = 31
A-C-B-E-D-A
6+8+6+8+8 = 36
A-C-D-B-E-A
6+9+5+6+4 = 30
A-C-E-D-B-A
6+7+8+5+7 = 33
A-D-B-C-E-A
8+5+8+9+4 = 34
A-B-E-C-D-A
7+6+7+9+8 = 37
The minimum total distance traveled is 30 and belongs to the Hamiltoncircuit A-C-D-B-E-A.
4. Apply Kruskal’s Algorithm to determine a minimum spanning tree ineach graph.
a.
b.