Presentation on theme: "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."— 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?
Theorem The player who makes the first move wins Bridg-it. Proof (Strategy stealing) Suppose Player 2 has a winning strategy. Player 1s 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 – Random Graphs Threshold Functions (First & Second Moment Method, Occurences of Subgraphs) Sharp Result for Connectivity Probabilistsic Method Chromatic Number and Jansons Inequalities The Phase Transition
Orga Exam – 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