Bridg-it by David Gale.

Slides:

Advertisements

Similar presentations

MCA 202: Discrete Mathematics under construction Instructor Neelima Gupta

Advertisements

Great Theoretical Ideas in Computer Science for Some.

CompSci 102 Discrete Math for Computer Science April 19, 2012 Prof. Rodger Lecture adapted from Bruce Maggs/Lecture developed at Carnegie Mellon, primarily.

Applied Combinatorics, 4th Ed. Alan Tucker

Great Theoretical Ideas in Computer Science.

Last time: terminology reminder w Simple graph Vertex = node Edge Degree Weight Neighbours Complete Dual Bipartite Planar Cycle Tree Path Circuit Components.

Graph Theory Ming-Jer Tsai. Outline Text Book Graph Graph Theory - Course Description The Topics in the Class Evaluation.

Prof. Amr Goneid, AUC1 Analysis & Design of Algorithms (CSCE 321) Prof. Amr Goneid Department of Computer Science, AUC Part R5. Graphs.

Online Ramsey Games in Random Graphs Reto Spöhel Joint work with Martin Marciniszyn and Angelika Steger.

What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.

Graph Theory Ming-Jer Tsai. Outline Graph Graph Theory Grades Q & A.

Great Theoretical Ideas in Computer Science.

Tucker, Applied Combinatorics, Section 1.4, prepared by Patti Bodkin

Graph Colouring Lecture 20: Nov 25.

Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.

Online Vertex Colorings of Random Graphs Without Monochromatic Subgraphs Reto Spöhel, ETH Zurich Joint work with Martin Marciniszyn.

Drawing of G. Planar Embedding of G Chord A chord of a cycle C is an edge not in C whose endpoints lie in C.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 11, Wednesday, September 24.

Drawing of G. Planar Embedding of G Proposition Proof. 1. Consider a drawing of K 5 or K 3,3 in the plane. Let C be a spanning cycle. 2. If the.

© 2007 Ray S. Babcock Tracks Game is played on a (nxn) set of squares. There are three possible moves (labeled A,B,C). Players alternate making a move.

Lecture 20: April 12 Introduction to Randomized Algorithms and the Probabilistic Method.

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 10, Monday, September 22.

Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger.

Introduction to Graph Theory

Graph Theory Topics to be covered:

Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.

Online Vertex-Coloring Games in Random Graphs Reto Spöhel (joint work with Martin Marciniszyn; appeared at SODA ’07)

The Tutte Polynomial Graph Polynomials winter 05/06.

 2. Region(face) colourings  Definitions 46: A edge of the graph is called a bridge, if the edge is not in any circuit. A connected planar graph is called.

V Spanning Trees Spanning Trees v Minimum Spanning Trees Minimum Spanning Trees v Kruskal’s Algorithm v Example Example v Planar Graphs Planar Graphs v.

Lines in the plane, slopes, and Euler’s formula by Tal Harel

The countable character of uncountable graphs François Laviolette Barbados 2003.

Planar Graphs. A graph is called planar if it can be drawn in the plane in such a way that no two edges cross. Example of a planar graph: The clique on.

MAT 2720 Discrete Mathematics Section 8.7 Planar Graphs

Planar Graphs Graph Coloring

Chapter 10 Graph Theory Eulerian Cycle and the property of graph theory 10.3 The important property of graph theory and its representation 10.4.

Some NP-complete Problems in Graph Theory Prof. Sin-Min Lee.

Graph Colouring Lecture 20: Nov 25. This Lecture Graph coloring is another important problem in graph theory. It also has many applications, including.

Graph Theory and Applications

Planar Graphs Lecture 10: Oct 21. This Lecture Today we will talk about planar graphs, and how to color a map using 6 colors. Planar graphs Euler’s formula.

Trees Thm 2.1. (Cayley 1889) There are nn-2 different labeled trees

Graph Theory Ming-Jer Tsai. Outline Graph Graph Theory Grades Q & A.

Great Theoretical Ideas in Computer Science for Some.

Chromatic Coloring with a Maximum Color Class Bor-Liang Chen Kuo-Ching Huang Chih-Hung Yen* 30 July, 2009.

COMPSCI 102 Introduction to Discrete Mathematics.

What is the chromatic number of this graph?. We have found a four-coloring. How do we know the chromatic number is not less than four?

Hex: a Game of Connecting Faces. Player 1 Player 2 Players take turns placing blue chips (player 1) and red chips (player 2). Player 1 plays first. Player.

Graphs. Representations of graphs ： undirected graph An undirected graph G have five vertices and seven edges An adjacency-list representation of G The.

Review: Discrete Mathematics and Its Applications

Outline 1 Properties of Planar Graphs 5/4/2018.

Discrete Mathematics Graph: Planar Graph Yuan Luo

Characteristics of Planar Graphs

The countable character of uncountable graphs François Laviolette Barbados 2003.

Graph theory Definitions Trees, cycles, directed graphs.

Great Theoretical Ideas in Computer Science

Chapter 5. Optimal Matchings

Tucker, Applied Combinatorics, Sec 2.4

Planarity Testing.

Discrete Mathematics for Computer Science

Lecture 15: Graph Theory II

Discrete Math II Howon Kim

Review: Discrete Mathematics and Its Applications

MAT 2720 Discrete Mathematics

Existence of 3-factors in Star-free Graphs with High Connectivity

N(S) ={vV|uS,{u,v}E(G)}

Gaph Theory Planar Graphs

Discrete Mathematics for Computer Science

GRAPH THEORY Properties of Planar Graphs Ch9-1.

GRAPH THEORY Properties of Planar Graphs Ch9-1.

Presentation transcript:

Bridg-it by David Gale

Bridg-it on Graphs Two players and alternately claim edges from the blue and the red lattice respectively. Edges must not cross. Objective: build a bridge 1: connect left and right 2: connect bottom and top Who wins Bridg-it?

Who wins Bridg-it? Theorem The player who makes the first move wins Bridg-it. Proof (Strategy stealing) Suppose Player 2 has a winning strategy. Player 1’s first move is arbitrary. Then Player 1 pretends to be Player 2 by playing his strategy. (Note: here we use that the field is symmetric!) Hence, Player 1 wins, which contradicts our assumption.

How does Player 1 win?

The Tool for Player 1 Proposition Suppose T and T’ are spanning trees of a connected graph G and e 2 E(T) n E(T’). Then there exists an edge e’ 2 E(T’) n E(T) such that T – e + e’ is a spanning tree of G.

Contents - Graphs Connected Graphs Eulerian/Hamiltonian Graphs Trees (Characterizations, Cayley‘s Thm, Prüfer Code, Spanning Trees, Matrix-Tree Theorem) k-connected Graphs (Menger‘s Thm, Ears Decomposition, Block-Decomposition, Tutte‘s Thm for 3-connected) Matchings (Hall‘s Thm, Tutte‘s Thm) Planare Graphs (Euler‘s Formula, Number Edges, Maximal Graphs) Colorings (Greedy, Brook‘s Thm, Vizing‘s Thm)

Contents – Random Graphs Threshold Functions (First & Second Moment Method, Occurences of Subgraphs) Sharp Result for Connectivity Probabilistsic Method Chromatic Number and Janson‘s Inequalities The Phase Transition

Orga Exam Challenge I: winner will be announced on website Freitag, 26. Juli, 14-16, B 051 Open Book Keine elektronische Hilfsmittel (Handy etc.) Challenge I: winner will be announced on website Challenge II: will be released in the week after the exam