Ramsey numbers of cycles under Gallai colorings

Date:2020-01-07Clicks:10设置

Speaker: Professor Song Zixia

Topic: Ramsey numbers of cycles under Gallai colorings

Date: Jan 7

Time: 14:00 pm

Venue: Lecture Hall 1508, Building 9

Sponsor: School of Mathematics and Statistics, Institute of Science and Technology

Abstract:

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 amonochromatic 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 Dayand 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 evencycles under Gallai coloring.

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


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