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{\bf David Rolnick }
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{\bf Trees with an On-Line Degree Ramsey Number of Four}
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On-line Ramsey theory studies a graph-building game between two players. The player called Builder builds edges 
one at a time, and the player called Painter paints each new edge red or blue after it is built. The graph 
constructed is called the \emph{background graph}. Builder's goal is to cause the background graph to contain a 
monochromatic copy of a given \emph{goal graph}, and Painter's goal is to prevent this. In the 
\emph{$S_k$-game} variant of the typical game, the background graph is constrained to have maximum degree no 
greater than $k$. The on-line degree Ramsey number $\mathring{R}_{\Delta}(G)$ of a graph $G$ is the minimum $k$ 
such that Builder 
wins an $S_k$-game in which $G$ is the goal graph. Butterfield et al. previously determined all graphs $G$ 
satisfying $\mathring{R}_{\Delta}(G)\le 3$. We provide a complete classification of trees $T$ satisfying 
$\mathring{R}_{\Delta}(T)=4$.

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