## Ramsey and star-critical Ramsey numbers involving generalized fans

- Sept. 6, 2019
- 2:30 p.m.
- LeConte 312

## Abstract

For graphs $G$ and $H$, the *Ramsey number* of $G$ and $H$ is the smallest number of vertices in a complete graph for which any red/blue edge-coloring contains a red $G$ or a blue $H$. Recently the variant called *star-critical Ramsey number* was introduced and it is a slightly more precise measure on when the “unavoidability” property occurs for graphs $G$ and $H$. More precisely, if $R(G,H)=k$, then take $\Gamma _ m$ to be the graph $K _ {k-1}$ along with an additional vertex of degree $m$; then $r _ * (G,H)$ is the smallest $m$ for which any red/blue edge-coloring of $\Gamma _ m$ contains a red $G$ or a blue $H$.

A generalized fan is a graph formed by taking some number of disjoint copies of a graph $H$ and joining each to a single vertex. Recently, there has been an increased interest in Ramsey and star-critical Ramsey numbers involving this kind of graph which motivated our work. In this talk I will discuss Ramsey and star-critical Ramsey numbers of generalized fans vs. complete graphs and of generalized fans vs. a disjoint union of triangles. Joint work with Paul Hazelton and Suzanna Thompson.