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arXiv · 1710.10455

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

Haibo Wu·Colton Magnant·Pouria Salehi Nowbandegani·Suman Xia
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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.

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Haibo Wu, Colton Magnant, Pouria Salehi Nowbandegani, Suman Xia. 2017-10-28. All partitions have small parts - Gallai-Ramsey numbers of bipartite graphs. https://arxiv.org/abs/1710.10455

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