Graph Theory

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# Graph Theory - PowerPoint PPT Presentation

Graph Theory. Ming-Jer Tsai. Outline. Text Book Graph Graph Theory - Course Description The Topics in the Class Evaluation. Text Books. &quot;Introduction to Graph Theory&quot;, 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.

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### 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

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
• 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 Topicsin the Class
• Matching
• Connectivity
• Coloring
• Planar Graphs
• Hamiltonian Cycles
Matching
• Matching: A matching in a graph G is a set of non-loop edges with no shared endpoints
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

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
• 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.

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.

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
• (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
• (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) Every planar graph is 4-colorable.
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
• 2 Mid-term Exams (40%)
• 1 Final (25%)
• 10 Quizzes (20%)
• Discussion (15%)