On online Ramsey theory

Abstract

The investigation of online Ramsey theory on specific graph classes was initiated in 2004 by Grytczuk, Hal uszczak, and Kierstead. They studied online Ramsey theory on forests, k-colorable graphs, outerplanar graphs, and planar graphs. Recently, Petrickova continued the study of online Ramsey theory on planar graphs further by disproving a conjecture by Grytczuk, Hal uszczak, and Kierstead. We carry on the study of online Ramsey theory by considering classes of graphs with one forbidden subgraph. We focus on characterizing the classes where Painter wins the online Ramsey game for $C_3$ on C where C is the class of F-free graphs for a connected graph F

we succeed the characterization except for when F is one particular graph. We also show that Painter wins the online Ramsey game for $C_3$ on C when C is the class of $K_4$ -minor-free graphs, extending a result by Grytczuk, Haluszczak, and Kierstead.