Size Ramsey Number of Bounded Degree Graphs for Games

Abstract

We study Maker/Breaker games on the edges ofsparsegraphs. Maker and Breaker take turns at claiming previously unclaimed edges of a given graphH. Maker aims to occupy a given target graphGand Breaker tries to prevent Maker from achieving his goal. We show that for everydthere is a constantc=c(d)with the property that for every graphGonnvertices of maximum degreedthere is a graphHon at mostcnedges such that Maker has a strategy to occupy a copy ofGin the game onH.This is a result about a game-theoretic variant of the size Ramsey number. For a given graphG,$\hat{r}'(G)$is defined as the smallest numberMfor which there exists a graphHwithMedges such that Maker has a strategy to occupy a copy ofGin the game onH. In this language, our result yields that for every connected graphGof constant maximum degree,$\hat{r}'(G) = \Theta(n)$.Moreover, we can also use our method to settle the corresponding extremal number foruniversalgraphs: for a constantdand for the class${\cal G}_{n}$ofn-vertex graphs of maximum degreed,$s({\cal G}_{n})$denotes the minimum number such that there exists a graphHwithMedges where, foreveryG∈${\cal G}_{n}$, Maker has a strategy to build a copy ofGin the game onH. We obtain that$s({\cal G}_{n}) = \Theta(n^{2 - \frac{2}{d}})$.