1 / 17

Graph Theory

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.

mirra
Download Presentation

Graph Theory

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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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. 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.

  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 Topicsin 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. 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

  9. 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

  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. (G) = 1. ’(G) = 2. (G) = 3.

  11. 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.

  12. 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.

  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 K5 or K3,3.

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

  16. 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.

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

More Related