Polytechnic University of the Philippines
College of Communication
Bayog, Anthony James P.
BABR 1-2
Mathematics in the Modern World
Sir Edcon B. Baccay
Lesson 6 Assessment: Mathematics of Graphs
1. Let V = {cities of Metro Manila} and E = {(x, y) | x and y are adjacent cities of Metro
Manila}.
a. Draw the graph G defined by G = (V, E). You may use initials to name a vertex
representing a city.
Let
V = {Nc, V, MB, C, QC, MR, NV, MN, SJ, MD, PG, PT, TG, MK, PY, PN, LP, MT}
E = {(x, y) | x and y are adjacent cities of Metro Manila}
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Polytechnic University of the Philippines
College of Communication
b. Apply the Four-Color Theorem to determine the chromatic number of the
vertex coloring for G.
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Polytechnic University of the Philippines
College of Communication
2. Apply Euler’s Theorems and Fleury’s Algorithm to determine Euler path and Euler
circuits in each graph.
a.
Euler’s Theorem 1: If a
graph is connected and
every vertex has an even
degree, then it has at least
one Euler circuit. If a
graph has at least one
Euler circuit, then it must
have a Euler path.
Graph (a) has a Euler
circuit and a Euler path.
Euler path and circuit: AG-B-H-G-F-E-H-C-J-E-D-C-B-A
b.
Euler’s Theorem
2: If a graph is
connected and has
just two vertices of
odd degree, then it
has at least one
Euler path. Any
path must start at
one of the odddegree vertices
and end at the
other one.
Euler path: H-D-C-H-L-M-H-G-L-K-E-J-K-F-G-C-B-F-E-A-B-E
3. A businessman had to visit five cities A, B, C, D, and E. The distance (in hundred
miles) between the five cities is as follows:
A
B
C
D
E
A
*
7
6
8
4
B
7
*
8
5
6
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Euler Circuit
Distance Travelled
A-B-C-D-E-A
36
A-B-C-E-D-A
38
C 6 8 * 9 7
A-D-C-B-E-A
35
D 8 5 9 * 8
A-B-E-D-C-A
36
A-C-B-D-E-A
31
A-C-B-E-D-A
36
A-C-D-B-E-A
30
A-C-E-D-B-A
33
A-D-B-C-E-A
32
A-B-E-C-D-A
36
Polytechnic University of the Philippines
College of32
Communication
A-B-D-C-E-A
E 4 6 7 8 *
If the businessman 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.
Answer: The businessman must
travel the path A-C-D-B-E-A to
minimize the total distance travelled.
4.
Apply Kruskal’s Algorithm to determine a
minimum spanning tree in each graph.
a.
Minimum spanning tree
Selected edges: {(b, c), (d, e), (b, d),
(a,b)} with total weight of 11
b.
Minimum spanning tree
Selected edges: {(d, e), (c, d), (e, f),
(b, e), (a, b), (c, g), (g, h), (e, i), (i,
j), (i, l), (g, k)} with total weight of 36
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