On-line Ramsey theory for bounded degree graphs

Abstract

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 R̊Δ(G) is the least k such that Builder wins (G,H) when H is the class of graphs with maximum degree at most k. Our results include: 1) R̊Δ(G)≤3 if and only if G is a linear forest or each component lies inside K1,3. 2) RΔ(G) ≥ Δ(G) + t - 1, where t = maxuv ∈E(G) min{d(u), d(v)}. 3) R̊Δ(G) ≤ d1+d2-1 for a tree G, where d1 and d2 are two largest vertex degrees. 4) 4 ≤ RΔ(Cn) ≤ 5, with R̊δ(Cn) = 4 except for finitely many odd values of n. 5) RΔ(G) ≤ 6 when Δ(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".