All partitions have small parts - Gallai-Ramsey numbers of bipartite graphs

TL;DR

This paper investigates Gallai-Ramsey numbers for bipartite graphs within Gallai-colorings, establishing that only 3-colorings with specific partitions need consideration, and determines these numbers for bipartite graphs with two vertices in one part.

Contribution

It introduces a reduction to 3-colorings with special partitions for bipartite graphs and determines Gallai-Ramsey numbers for all bipartite graphs with two vertices in one part.

Findings

Gallai-Ramsey numbers are determined for bipartite graphs with two vertices in one part.

A reduction to 3-colorings with specific partitions simplifies the problem.

Conjectures are made for the sharp values of all 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 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 all bipartite graphs with two vertices in one part and initiate the study of linear forests.