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 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

Connected Graph connected graphdisconnected graph 11

Multigraph 12

Subgraph 13

Spanning Subgraph 14

Induced Subgraph 15

Which of the following is an induced subgraph of G? O Induced Subgraph O X 16

Components of a Graph 12 connected sugraph 17

G-vG-v 18

G-eG-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? O X 23

Isomorphic Graphs 24

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 25

Vertex Degree 26

Theorem

Corollary

29

a b c d 30

Euler Circuit and Euler Trail 31

Theorem 11.3 ( ⇒)

Theorem

Theorem 11.3 ( ⇐)

Theorem

Theorem

Corollary 11.2 ( ⇐) ( ⇒) 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? OO 40

Euler’s Theorem v = e = r = v – e + r =

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. –Therefore, v-e+r=2. e (induction hypothesis) 43

Corollary It suffices to consider connected graphs; otherwise, we could add edges. 2. If v  3, every region contains at least three edges (  L(R i )  3r). 3. 2e=  L(R i ), implying 2e  3r. 4. By Euler’s Theorem, v–e+r=2, implying e≤ 3v– 6. If also G is triangle-free, then e ≤ 2v–4. (  L(R i )  4r) (2e  4r) (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.) 44

Bipartite Graph 45

Nonplanarity of K 5 and K 3,3 K 5 (e = 10, n = 5) K 3,3 (e = 9, n = 6) These graphs have too many edges to be planar. –For K 5, we have e = 10>9 = 3n-6. –Since K 3,3 is triangle-free, we have e = 9>8 = 2n-4. 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

Theorem

Theorem

Theorem

Theorem

Theorem

Proper Coloring and Chromatic Number 59

Counting Proper Colors

61

Theorem

Example

Example