Component Games on Regular Graphs

Abstract

We study the (1:b) Maker–Breaker component game, played on the edge set of ad-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to prevent him from doing so. For all values of Breaker's biasb, we determine whether Breaker wins (on anyd-regular graph) or Maker wins (on almost everyd-regular graph) and provide explicit winning strategies for both players.To this end, we prove an extension of a theorem of Gallai, Hasse, Roy and Vitaver about graph orientations without long directed simple paths.