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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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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.

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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, 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)

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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 Janson‘s Inequalities The Phase Transition

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**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

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