Component Games on Regular Graphs

TL;DR

This paper analyzes the (1:b) Maker-Breaker component game on d-regular graphs, determining winning conditions for both players and providing strategies, extending a theorem on graph orientations without long directed paths.

Contribution

It establishes the winning conditions for Maker and Breaker on d-regular graphs and extends a theorem on graph orientations, offering explicit strategies.

Findings

Breaker's win for certain biases b on all d-regular graphs.

Maker's win on almost every d-regular graph for other biases.

Extension of Gallai-Hasse-Roy-Vitaver theorem to paths without long directed simple paths.

Abstract

We study the (1:b) Maker-Breaker component game, played on the edge set of a d-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to not let him do so. For all values of Breaker's bias b, we determine whether Breaker wins (on any d-regular graph) or Maker wins (on almost every d-regular graph) and provide explicit winning strategies for both players. To this end, we prove an extension of a theorem by Gallai-Hasse-Roy-Vitaver about graph orientations without long directed simple paths.