All partitions have small parts — Gallai–Ramsey numbers of bipartite graphs

Abstract

Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider monochromatic bipartite graphs since the numbers are known to grow more slowly than for non-bipartite graphs. The main result shows that it often suffices to consider only 3-colorings which have a special partition of the vertices. Using this tool, we find several sharp numbers and conjecture the sharp value for all bipartite graphs. In particular, we determine the Gallai–Ramsey numbers for many bipartite graphs with two vertices in one part and initiate the study of linear forests.