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?

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.

How does Player 1 win?

The Tool for Player 1

PropositionSuppose 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.

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)

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

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