On-line Ramsey Theory for Bounded Degree Graphs

When graph Ramsey theory is viewed as a game, “Painter” 2-colors the edges of a graph presented by “Builder”. Builder wins if every coloring has a monochromatic copy of a fixed graph G. In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class H. Builder wins the game (G,H) if a monochromatic copy of G can be forced. The on-line degree Ramsey number R̊∆(G) is the least k such that Builder wins (G,H) when H is the class of graphs with maximum degree at most k. Among our results are the following: 1) R̊∆(G)≤3 if and only if G is a linear forest or each component is contained in K1,3. 2) R̊∆(G) ≥ ∆(G) + t− 1, where t = maxuv∈E(G)min{d(u), d(v)}. 3) R̊∆(G) ≤ d1 + d2 − 1 for a tree G, where d1, . . . , dn is the nonincreasing degree list. 4) 4 ≤ R̊∆(Cn) ≤ 5, with R̊∆(Cn) = 4 except for finitely many odd values of n. 5) R̊∆(G) ≤ 6 when ∆(G) ≤ 2. The lower bounds come from strategies for Painter that color edges red whenever the red graph remains in a specified class. The upper bounds use a result showing that Builder may assume that Painter plays “consistently”.