1. Graph Theory Ming-Jer Tsai

2. Outline Text Book Graph Graph Theory - Course Description The Topics in the Class Evaluation

3. Text Books "Introduction to Graph Theory", Douglas B. West, 2rd Edition, Prentice Hall

4. 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. e1e1 e2e2 e6e6 e5e5 e3e3 e7e7 e4e4 x yw z

5. 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.

6. The Topics in the Class Matching Connectivity Coloring Planar Graphs Hamiltonian Cycles

7. Matching Matching: A matching in a graph G is a set of non-loop edges with no shared endpoints

8. (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. Matching BCDEA X Y S = {B, D, E}

9. (Tutte’s Condition) A graph G has a perfect matching iff o(G-S)<=|S| for every S  V(G). o(G-S): the number of components of odd orders in G-S. Matching S Odd component Even component

10. 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. 1.  (G) = 1. 2.  ’(G) = 2. 3.  (G) = 3.

11. ( 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. Connectivity

12. (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. Coloring 1 2 3 4 5 6 1 2 3 4 5 6 1 2 4 3

13. 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.

14. Planar Graph (Kuratowski’s Theorem) A graph is planar iff it does not contain a subdivision of K 5 or K 3,3.

15. (Four Color Theorem) Every planar graph is 4- colorable. Four Color Theorem

16. 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 d 1 ≤ … ≤ d n, where n ≥ 3. If i i or d n-i ≥ n-i, G has a hamiltonian cycle. Hamiltonian Cycles

17. Evaluation 2 Mid-term Exams (40%) 1 Final (25%) 10 Quizzes (20%) Discussion (15%)