Redundant edges in Ramsey graphs

TL;DR

This paper investigates the role of redundant edges in Ramsey graphs, defining and determining critical Ramsey numbers for various classes of graphs such as stars, matchings, and complete graphs.

Contribution

It introduces the concept of critical Ramsey numbers relative to classes of graphs and computes these numbers for specific graph classes, advancing understanding of edge redundancy in Ramsey theory.

Findings

Determined critical Ramsey numbers for star graphs.

Computed critical Ramsey numbers for matchings.

Analyzed critical Ramsey numbers for complete graphs.

Abstract

For graphs $G$, $F$ and $H$, let $G\rightarrow (F,H)$ signify that any edge coloring of $G$ in red and blue contains a red $F$ or a blue $H$. The Ramsey number $R(F,H)=\min\{r|\; K_r\rightarrow (F,H)\}$. In this note, we consider redundant edges in Ramsey graphs, which are associate with critical Ramsey numbers. For an integer $k\ge 1$, let ${\mathbb G}=\{G_k,G_{k+1},\dots \}$ be a class of graphs with $\delta(G_n)\ge 1$. We define the critical Ramsey number $R_{\mathbb G}(F,H)$ with respect to $\mathbb G$ to be $\max\{n|\; K_r\setminus G_n \rightarrow(F,H),\,G_n\in{\mathbb G}\big\}$, where $r=R(F,H)$. We shall determine some $R_{\mathbb G}(F,H)$, where ${\mathbb G}$ consists of stars, matchings and complete graphs, respectively.