Two types of size Ramsey numbers for matchings of small order

TL;DR

This paper investigates the size Ramsey numbers and connected size Ramsey numbers for specific small graphs, providing new bounds and exploring their relationships for various graph classes.

Contribution

It introduces new results on the values of size Ramsey numbers for matchings and cycles, and improves bounds for certain cases involving matchings and paths.

Findings

Determined values of size Ramsey numbers for certain small graphs.

Established relationships between size Ramsey numbers and their connected variants.

Improved upper bounds for specific cases involving matchings and paths.

Abstract

For simple graphs $G$ and $H$, their size Ramsey number $\hat{r}(G,H)$ is the smallest possible size of $F$ such that for any red-blue coloring of its edges, $F$ contains either a red $G$ or a blue $H$. Similarly, we can define the connected size Ramsey number ${\hat{r}}_c(G,H)$ by adding the prerequisite that $F$ must be connected. In this paper, we explore the relationships between these size Ramsey numbers and give some results on their values for certain classes of graphs. We are mainly interested in the cases where $G$ is either a $2K_2$ or a $3K_2$, and where $H$ is either a cycle $C_n$ or a union of paths $nP_m$. Additionally, we improve an upper bound regarding the values of $\hat{r}(tK_2,P_m)$ and ${\hat{r}}_c(tK_2,P_m)$ for certain $t$ and $m$.