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Bridg-it by David Gale

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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?

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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.

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How does Player 1 win?

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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.

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Contents - Graphs Connected Graphs Eulerian/Hamiltonian Graphs Trees (Characterizations, Cayleys Thm, Prüfer Code, Spanning Trees, Matrix-Tree Theorem) k-connected Graphs (Mengers Thm, Ears Decomposition, Block-Decomposition, Tuttes Thm for 3-connected) Matchings (Halls Thm, Tuttes Thm) Planare Graphs (Eulers Formula, Number Edges, Maximal Graphs) Colorings (Greedy, Brooks Thm, Vizings Thm)

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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

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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

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