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?
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.

4. How does Player 1 win?

5. 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.

6. 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)

7. 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

8. 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