Generating random graphs in biased Maker-Breaker games

TL;DR

This paper introduces a new approach linking biased Maker-Breaker games with local resilience in random graphs, leading to novel results about graph properties Maker can achieve under certain biases.

Contribution

It develops a general method connecting biased Maker-Breaker games to local resilience problems, deriving new and known results about graph structures Maker can construct.

Findings

Maker can build a pancyclic graph for b=o(√n)

Maker can embed all bounded-degree spanning trees with a linear bare path for b=Θ(n/ln n)

The approach unifies and extends previous results in biased Maker-Breaker games.

Abstract

We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for $b=o\left(\sqrt{n}\right)$, Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a $(1:b)$ game on $E(K_n)$. As another application, we show that for $b=\Theta\left(n/\ln n\right)$, playing a $(1:b)$ game on $E(K_n)$, Maker can build a graph which contains copies of all spanning trees having maximum degree $\Delta=O(1)$ with a bare path of linear length (a bare path in a tree $T$ is a path with all interior vertices of degree exactly two in $T$).