Chap. 11 Graph Theory and
Applications
1
Directed Graph
2
(Undirected) Graph
3
Vertex and Edge Sets
4
Walk
5
Closed (Open) Walk
6
Trail, Path, Circuit, and Cycle
7
Comparison of Walk, Trail, Path,
Circuit, and Cycle
8
Theorem 11.1
Observation:
9
Theorem 11.1
1. It suffices to show from a to b,
the shortest trail is the shortest path.
2. Let
be the shortest trail
from a to b.
3.
4.
10
Connected Graph
connected graph
disconnected graph
11
Multigraph
12
Subgraph
13
Spanning Subgraph
14
Induced Subgraph
15
Induced Subgraph
Which of the following is an induced subgraph of G?
O
O
X
16
Components of a Graph
connected sugraph
1
2
17
G-v
18
G-e
19
Complete Graph
20
Complement of a Graph
21
Isomorphic Graphs
22
Isomorphic Graphs
Which of the following function define a graph
isomorphism for the graphs shown below?
X
O
23
Isomorphic Graphs
24
Isomorphic Graphs
Are the following two graphs isomorphic? X
In (a), a and d each adjacent to two other vertices.
25
In (b), u, x, and z each adjacent to two other vertices.
Vertex Degree
26
Theorem 11.2
27
Corollary 11.1
28
29
a
b
d
c
30
Euler Circuit and Euler Trail
31
Theorem 11.3
(⇒) 1.
2.
3.
4.
5.
6.
7.
32
Theorem 11.3
8.
9.
33
Theorem 11.3
(⇐) 1.
2.
3.
34
Theorem 11.3
4.
5.
6.
7.
8.
35
Theorem 11.3
9.
10.
11.
12.
13.
14.
36
Corollary 11.2
(⇐) 1.
2.
3.
4.
(⇒) The proof of only if part is similar to that of Theorem
11.3 and omitted.
37
Incoming and Outgoing Degrees
2
38
Theorem 11.4
The proof is similar to that of Theorem 11.3
and omitted.
39
Planar Graph
Which of the following is a planar graph?
O
O
40
Euler’s Theorem
v=7
e= 8
r= 3
v–e+r=2
41
Euler’s Theorem
• Proof. 1. Use induction on v (number of vertices).
• 2. Basis (v = 1):
– G is a “bouquet” of loops, each a closed curve in
the embedding.
– If e = 0, then r = 1, and the formula holds.
– Each added loop passes through a region and cuts
it into 2 regions. This augments the edge count and
the region count each by 1. Thus the formula holds
when v = 1 for any number of edges.
42
Euler’s Theorem
• 3. Induction step (v>1):
– There exists an edge e that is not a loop
because G is connected.
– Obtain a graph G’ with v’ vertices, e’ edges, and r’
regions by contracting e.
– Clearly, v’=v–1, e’=e–1, and r’=r.
– v’– e’+ r’ = 2. (induction hypothesis)
– Therefore, v-e+r=2.
e
43
Corollary 11.3
If G is a simple planar graph with at least three vertices,
then e≤3v–6. (A simple graph is not a multigraph and does not contain any loop.)
If also G is triangle-free, then e ≤ 2v–4.
1. It suffices to consider connected graphs; otherwise, we
could add edges.
2. If v 3, every region contains at least three edges
(L(Ri) 3r). (L(Ri) 4r)
3. 2e=L(Ri), implying 2e3r. (2e4r)
4. By Euler’s Theorem, v–e+r=2, implying e≤ 3v– 6.
(e≤ 2v–4) 44
Bipartite Graph
45
Nonplanarity of K5 and K3,3
• These graphs have too many edges to be planar.
– For K5, we have e = 10>9 = 3n-6.
– Since K3,3 is triangle-free, we have e = 9>8 = 2n-4.
K5 (e = 10, n = 5)
K3,3 (e = 9, n = 6)
46
Subdivision of a Graph
47
Subdivision of a Graph
48
49
50
Hamilton Cycle
51
Hamilton Cycle
Does the following graph contain a hamiltion cycle? X
52
Theorem 11.8
1.
2.
3.
4.
53
Theorem 11.8
5.
6.
7.
8.
9.
10.
11.
12.
13.
54
Theorem 11.8
14.
15.
16.
17.
18.
19.
55
Theorem 11.8
17.
18.
19.
20.
56
Theorem 11.9
1.
2.
3.
4.
5.
57
Theorem 11.9
6.
7.
8.
9.
10.
11.
58
Proper Coloring and Chromatic
Number
59
Counting Proper Colors
1.
2.
3.
4.
60
61
Theorem 11.10
1.
2.
3.
4.
5.
6.
62
Example 11.36
63
Example 11.37
64