報 告 人︰ 宋梓霞 教授

報告題目︰Ramsey numbers of cycles under Gallai colorings





       宋梓霞博士是美國中佛羅里達大學(University of Central Florida)數學系教授,博士生導師。主要研究領域為圖論。宋梓霞博士于2000-2004年在美國佐治亞理工大學(Georgia Institute of Technology)獲算法,組合,優化(Algorithm, Combinatorics and Optimization)博士學位,2004-2005年在美國俄亥俄州立大學(The Ohio State University)數學系從事博士後研究。2005年授聘于美國中佛羅里達大學數學系。獲得2009-2011美國NSA科研基金,是美國自然科學基金(NSF)的基金評委。2013年獲校優秀教師獎。


       For a graph $H$ and an integer $k\geq1$, the $k$-color Ramsey number $R_{k}(H)$ is the least integer $N$ such that every $k$-coloring of the edges of the complete graph $K_{N}$ contains a monochromatic copy of $H$. Let $C_{m}$ denote the cycle on $m\geq4$ vertices. For odd cycle, Bondy and Erd\H{o}s in 1973 conjectured that for all $k\geq1$ and $n\geq2$, $R_{k}(C_{2n+1})=n2^{k}+1$.Recently, this conjecture has been verified to be true for all fixed $k$ and all $n$ sufficiently large by Jenssen and Skokan; and false for all fixed $n$ and all $k$ sufficiently large by Day and Johnson. Even cycles behave rather differently in this context. Little is known about the behavior of $R_{k}(C_{2n})$ is general. In this talk we will present our recent results on Ramsey numbers of cycles under Gallai coloring, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. We also completely determine the Ramsey number of even cycles under Gallai coloring.

Joint work with Dylan Bruce, Christian Bosse, Yaojun Chen and Fangfang Zhang.


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