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- W2071075508 abstract "We study Maker/Breaker games on the edges of sparse graphs. Maker and Breaker take turns at 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 show that for every d there is a constant c = c(d) with the property that for every graph G on n vertices of maximum degree d there is a graph H on at most cn edges such that Maker has a strategy to occupy a copy of G in the game on H . This is a result about a game-theoretic variant of the size Ramsey number. For a given graph G , $hat{r}'(G)$ is defined as the smallest number M for which there exists a graph H with M edges such that Maker has a strategy to occupy a copy of G in the game on H . In this language, our result yields that for every connected graph G of constant maximum degree, $hat{r}'(G) = Theta(n)$ . Moreover, we can also use our method to settle the corresponding extremal number for universal graphs: for a constant d and for the class ${cal G}_{n}$ of n -vertex graphs of maximum degree d , $s({cal G}_{n})$ denotes the minimum number such that there exists a graph H with M edges where, for every G ∈ ${cal G}_{n}$ , Maker has a strategy to build a copy of G in the game on H . We obtain that $s({cal G}_{n}) = Theta(n^{2 - frac{2}{d}})$ ." @default.
- W2071075508 created "2016-06-24" @default.
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- W2071075508 date "2013-06-11" @default.
- W2071075508 modified "2023-09-25" @default.
- W2071075508 title "Size Ramsey Number of Bounded Degree Graphs for Games" @default.
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- W2071075508 doi "https://doi.org/10.1017/s0963548313000151" @default.
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