Graph Theory Ming-Jer Tsai

Outline Graph Graph Theory Grades Q & A

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

Graph Theory 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

(Hall’s Condition) An X,Y-bigraph G has a matching that saturates X if and only if |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}

(Tutte’s Condition) A graph G has a perfect matching if and only if 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

( 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

(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

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 K 5 or K 3,3.

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

Textbook INTRODUCTION TO GRAPH THEORY, Douglas B. West, Prentice Hall ( 全華代理 )

Grades 4 Reading Reports (50%) 3 Review Reports (30%) Discussion and Attendance (20%) Bonus: Presentation (10%)

Reading Reports 1 st Topics: (Due 10/17)  Enumeration of Trees (in Sec. 2.2)  Spanning Trees in Graphs (in Sec. 2.2)  Maximum Bipartite Matching (in Sec. 3.2)  Weighted Bipartite Matching (in Sec. 3.2) 2 nd Topics: (Due 11/7)  Counting Proper Colorings (in Sec. 5.3)  Coloring of Planar Graphs (in Sec. 6.3)  Crossing Number (in Sec.6.3)  Sufficient Conditions for Hamilton Cycles (in Sec. 7.2)

Reading Reports 3 rd Topics: (Due 11/28)  Perfect Graphs (in Sec. 8.1)  Matroids (in Sec. 8.2)  Ramsey Theory (in Sec. 8.3)  More External Problems (in Sec. 8.4)  Random Graphs (in Sec. 8.5)  Eigenvalues of Graphs (in Sec. 8.6) 4 th Topics: (Due 12/26)  All the other topics introduced in the class.

Review Reports Each student reviews the reading reports of the other 3 students for each review report. 1 st : due 10/31. 2 nd : due 11/21. 3 rd : due 12/12.

Presentation For each topic, the one with the highest grade presents the topic. Each student presents at most one topic. 1 st : 11/28, 12/5. (Announce: 11/7) 2 nd : 12/12, 12/19. (Announce: 11/28) 3 rd : 12/26, 1/2, 1/9. (Announce: 12/19)

Remarks 10% algorithm + 90% theory in the class This course has nothing to do with computer graphics Prerequisite: Discrete Mathematics and Algorithm

Q & A