Gallai-colorings are edge-colored complete graphs in which there are norainbow triangles. Within such colored complete graphs, we consider Ramsey-typequestions, looking for specified monochromatic graphs. In this work, weconsider monochromatic bipartite graphs since the numbers are known to growmore slowly than for non-bipartite graphs. The main result shows that itsuffices to consider only $3$-colorings which have a special partition of thevertices. Using this tool, we find several sharp numbers and conjecture thesharp value for all bipartite graphs. In particular, we determine theGallai-Ramsey numbers for all bipartite graphs with two vertices in one partand initiate the study of linear forests.
展开▼
