Maker Can Construct a Sparse Graph on a Small Board

TL;DR

This paper investigates Maker/Breaker games on sparse graphs, demonstrating that Maker can secure a target graph G within a carefully constructed sparse graph H, depending on the degree d of G.

Contribution

It introduces a method to construct sparse graphs H that enable Maker to win on any d-regular target graph G, advancing understanding of positional games on sparse structures.

Findings

Maker can construct a copy of G on a sparse graph H.

The size of H depends linearly on the number of vertices and a function of d.

The result applies to all d-regular graphs G.

Abstract

We study Maker/Breaker games on the edges of sparse graphs. Maker and Breaker take turns in claiming previously unclaimed edges of a given graph H. Maker aims to occupy a given target graph G and Breaker tries to prevent Maker from achieving his goal. We define a function f and show that for every d-regular graph G on n vertices there is a graph H with at most f(d)n edges such that Maker can occupy a copy of G in the game on H.