On-line Ramsey Theory for Bounded Degree Graphs

Jane Butterfield, Tracy Grauman, William B. Kinnersley, Kevin G. Milans, Christopher Stocker, Douglas B. West
2011 Electronic Journal of Combinatorics  
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 ${\cal H}$. Builder wins the game $(G,{\cal H})$ if a monochromatic copy of $G$ can be forced. The on-line degree Ramsey number $\mathring {R}_\Delta(G)$ is the least $k$
more » ... ch that Builder wins $(G,{\cal H})$ when ${\mathcal H}$ is the class of graphs with maximum degree at most $k$. Our results include: 1) $\mathring {R}_\Delta(G)\!\le\!3$ if and only if $G$ is a linear forest or each component lies inside $K_{1,3}$. 2) $\mathring {R}_\Delta(G)\ge \Delta(G)+t-1$, where $t=\max_{uv\in E(G)}\min\{d(u),d(v)\}$. 3) $\mathring {R}_\Delta(G)\le d_1+d_2-1$ for a tree $G$, where $d_1$ and $d_2$ are two largest vertex degrees. 4) $4\le \mathring {R}_\Delta(C_n)\le 5$, with $\mathring {R}_\Delta(C_n)=4$ except for finitely many odd values of $n$. 5) $\mathring {R}_\Delta(G)\le6$ when $\Delta(G)\le 2$. 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".
doi:10.37236/623 fatcat:rx6c6374j5h4rgwhtnaqy6pa6m