2Bridg-it on GraphsTwo players and alternately claim edges from the blue and the red lattice respectively.Edges must not cross.Objective: build a bridge1: connect left and right2: connect bottom and topWho wins Bridg-it?
3Who 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.
5The Tool for Player 1Proposition 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.
7Contents – Random Graphs Threshold Functions (First & Second Moment Method, Occurences of Subgraphs)Sharp Result for ConnectivityProbabilistsic MethodChromatic Number and Janson‘s InequalitiesThe Phase Transition
8Orga Exam Challenge I: winner will be announced on website Freitag, 26. Juli, 14-16, B 051Open BookKeine elektronische Hilfsmittel (Handy etc.)Challenge I: winner will be announced on websiteChallenge II: will be released in the week after the exam