Building spanning trees quickly in Maker-Breaker games

TL;DR

This paper investigates the Maker-Breaker game on complete graphs, showing Maker can quickly build certain trees within n+1 moves, and almost all trees within n-1 moves, with some exceptions.

Contribution

It proves Maker can construct any bounded-degree tree in n+1 moves and almost all trees in n-1 moves, advancing understanding of rapid tree-building strategies in Maker-Breaker games.

Findings

Maker can win with bounded-degree trees in n+1 moves

Maker can build almost all trees in n-1 moves

Some tree families cannot be built in n-1 moves

Abstract

For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T. We prove that if T has bounded maximum degree, then Maker can win this game within n+1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n-1 moves and provide non-trivial examples of families of trees which Maker cannot build in n-1 moves.