graph theory n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Graph Theory PowerPoint Presentation
Download Presentation
Graph Theory

Loading in 2 Seconds...

play fullscreen
1 / 17

Graph Theory - PowerPoint PPT Presentation


  • 401 Views
  • Uploaded on

Graph Theory. Ming-Jer Tsai. Outline. Text Book Graph Graph Theory - Course Description The Topics in the Class Evaluation. Text Books. "Introduction to Graph Theory", Douglas B. West, 2rd Edition, Prentice Hall. x. e 1. e 6. e 2. w. y. e 5. e 4. e 7. e 3. z. Graph.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Graph Theory' - mirra


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
graph theory

Graph Theory

Ming-Jer Tsai

outline
Outline
  • Text Book
  • Graph
  • Graph Theory - Course Description
  • The Topics in the Class
  • Evaluation
text books
Text Books
  • "Introduction to Graph Theory", Douglas B. West, 2rd Edition, Prentice Hall
graph

x

e1

e6

e2

w

y

e5

e4

e7

e3

z

Graph
  • A triple consisting of a vertex set V(G), an edge set E(G), and a relation that associates with each edge two vertices (not necessary distinct) called its endpoints.
graph theory course description
Graph Theory - Course Description
  • The focus is on understanding the structure of graphs and exploring the proof techniques in discrete mathematics.
  • Students that would like to take this course are assumed to be interested in and have knowledge of discrete mathematics.
the topics in the class
The Topicsin the Class
  • Matching
  • Connectivity
  • Coloring
  • Planar Graphs
  • Hamiltonian Cycles
matching
Matching
  • Matching: A matching in a graph G is a set of non-loop edges with no shared endpoints
matching1
Matching
  • (Hall’s Condition) An X,Y-bigraph G has a matching that saturates X iff|N(S)|>=|S| for all SX.

N(S): the set of vertices having a neighbor in S.

S = {B, D, E}

A

B

C

D

E

X

Y

matching2
Matching
  • (Tutte’s Condition) A graph G has a perfect matching iffo(G-S)<=|S| for every SV(G).

o(G-S): the number of components of odd orders in G-S.

Odd component

Even component

S

connectivity
Connectivity
  • For a simple graph G, (G)<=’(G)<= (G).

(G): the minimum size of a vertex set S such that G-S is disconnected or has only one vertex (connectivity of G).

’(G): the minimum size of a set of edges F such that G-F has more than one component (edge-connectivity of G).

(G): minimum degree of G.

(G) = 1.

’(G) = 2.

(G) = 3.

connectivity1
Connectivity
  • (Menger Theorem) If x,y are vertices of a graph G and xyE(G), (x,y) = (x,y).

(x,y): the minimum size of a set SV(G)-{x,y} such that G-S has no x,y-path.

(x,y): the maximum size of a set of pairwise internally disjoint x,y-paths.

coloring

3

2

1

4

2

5

3

2

3

5

1

4

6

4

6

1

Coloring
  • (Brook’s Theorem) If G is a connected graph other than a complete graph or an odd cycle, (G)<=(G).

(G): The least k such that G is k-colorable.

(G): the maximum degree in G.

edge coloring
Edge-Coloring
  • (Vizing and Gupta’s Theorem) If G is a simple graph, x’(G) ≤Δ(G)+1.

’(G): The least k such that G is k-edge-colorable.

planar graph
Planar Graph
  • (Kuratowski’s Theorem) A graph is planar iff it does not contain a subdivision of K5 or K3,3.
four color theorem
Four Color Theorem
  • (Four Color Theorem) Every planar graph is 4-colorable.
hamiltonian cycles
Hamiltonian Cycles
  • If G is a simple graph with at least three vertices and δ(G) ≥ n(G)/2, then G has a hamiltonian cycle.
  • (Chvatal’s Condition) Let G be a simple graph with vertex degree d1 ≤ … ≤ dn, where n ≥ 3. If i < n/2 implies that di > i or dn-i ≥ n-i, G has a hamiltonian cycle.
evaluation
Evaluation
  • 2 Mid-term Exams (40%)
  • 1 Final (25%)
  • 10 Quizzes (20%)
  • Discussion (15%)