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Bridg -it by David Gale

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?.

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Bridg -it by David Gale

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  1. Bridg-it by David Gale

  2. 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?

  3. Who wins Bridg-it? TheoremThe 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.

  4. How does Player 1 win?

  5. The Tool for Player 1 PropositionSuppose T and T’ are spanning trees of a connected graph G and e2E(T) nE(T’). Then there exists an edge e’ 2E(T’) nE(T) such that T – e + e’ is a spanning tree of G.

  6. Contents - Graphs • Connected Graphs • Eulerian/Hamiltonian Graphs • Trees (Characterizations, Cayley‘sThm, Prüfer Code, SpanningTrees, Matrix-TreeTheorem) • k-connected Graphs (Menger‘sThm, EarsDecomposition, Block-Decomposition, Tutte‘sThmfor 3-connected) • Matchings (Hall‘sThm, Tutte‘sThm) • PlanareGraphs (Euler‘sFormula, NumberEdges, Maximal Graphs) • Colorings (Greedy, Brook‘sThm, Vizing‘sThm)

  7. Contents – Random Graphs • ThresholdFunctions (First & Second Moment Method, Occurencesof Subgraphs) • Sharp Resultfor Connectivity • ProbabilistsicMethod • ChromaticNumberandJanson‘sInequalities • The Phase Transition

  8. Orga • Exam • Freitag, 26. Juli, 14-16, B 051 • Open Book • Keine elektronische Hilfsmittel (Handy etc.) • Challenge I: winner will beannounced on website • Challenge II: will bereleased in theweek after theexam

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