Georgia Institute of Technology College of Sciences

A tournament is a directed graph obtained by orienting each edge of a complete graph. A set of vertices D is a dominating set in a tournament if every vertex not in D is pointed by a vertex in D. A tournament H is a rebel if there exists k such that every H-free tournament has a dominating set of size at most k. Wu conjectured that every tournament is a rebel. This conjecture, if true, implies several other conjectures about tournaments. However, we will prove that Wu's conjecture is false in general and prove a necessary condition for being rebels. In addition, we will prove that every 2-colorable tournament and at least one non-2-colorable tournament are rebels. The later implies an open case of a conjecture of Berger, Choromanski, Chudnovsky, Fox, Loebl, Scott, Seymour and Thomasse about coloring tournaments. This work is joint with Maria Chudnovsky, Ringi Kim, Paul Seymour and Stephan Thomasse.