Chap. 11 Graph Theory and Applications 1. Directed Graph 2

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:

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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 graph11

Multigraph

12

Subgraph

13

Spanning Subgraph

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Induced Subgraph

15

Which of the following is an induced subgraph of G? O

Induced Subgraph

O X

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Components of a Graph

1 2

connected sugraph

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G-v

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G-e

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Complete Graph

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Complement of a Graph

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Isomorphic Graphs

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Isomorphic Graphs

Which of the following function define a graph isomorphism for the graphs shown below?

OX

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Isomorphic Graphs

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Isomorphic Graphs

Are the following two graphs isomorphic?

In (a), a and d each adjacent to two other vertices.In (b), u, x, and z each adjacent to two other vertices.

X

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Vertex Degree

26

Theorem 11.2

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Corollary 11.1

28

29

a

b

c

d

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Euler Circuit and Euler Trail

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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.

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Incoming and Outgoing Degrees

2

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Theorem 11.4

The proof is similar to that of Theorem 11.3 and omitted.

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Planar Graph

Which of the following is a planar graph?

O O40

Euler’s Theorem

v =e =r =v – e + r = 2

783

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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.

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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. – Therefore, v-e+r=2.

e

(induction hypothesis)

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Corollary 11.3

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).

3. 2e=L(Ri), implying 2e3r.

4. By Euler’s Theorem, v–e+r=2, implying e≤ 3v– 6.

If also G is triangle-free, then e ≤ 2v–4.

(L(Ri) 4r)

(2e4r)

(e≤ 2v–4)

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.)

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Bipartite Graph

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Nonplanarity of K5 and K3,3

K5 (e = 10, n = 5) K3,3 (e = 9, n = 6)

• 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.

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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

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Theorem 11.8

1.2.

3.4.

53

Theorem 11.8

5.6.

7.8.9.10.11.12.

13.54

Theorem 11.814.15.

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.

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Example 11.36

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Example 11.37

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