Abstract
In the on-line Ramsey game on a family H of graphs, “Builder” presents edges of a graph one-by-one, and “Painter” colors each edge as it is presented; we require that Builder keep the presented graph in H. Builder wins the game (G, H) if Builder can ensure that a monochromatic G arises. The s-color on-line degree Ramsey number of G, denoted ˚ R¢(G; s), is the least k such that Builder wins (G, H) when H is the family of graphs having maximum degree at most k and Painter has s colors available. More generally, ˚ R¢(G1, . . . , Gs) is the minimum k such that Builder can force a copy of Gi in color i for some i when restricted to graphs with maximum degree at most k. In this paper, we prove that ˚ R¢(T ; s) ≤ s(¢( T ) − 1) + 1 for every tree T ; this is sharp, with equality whenever T has adjacent vertices of maximum degree. We also give lower and upper bounds on ˚ R¢(G1, . . . , Gs) when each Gi is a double-star. When each Gi is a star, we determine ˚ R¢(G1, . . . , Gs) exactly.
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