The relationship between some nonclassical Ramsey numbers

TL;DR

This paper investigates nonclassical Ramsey numbers related to domination and irredundant sets, proving key equalities, determining two unknown numbers, and solving four open cases in the field.

Contribution

It establishes the equality between two nonclassical Ramsey numbers, determines two previously unknown numbers, and resolves four open problems from prior research.

Findings

Proved that v(3,n)=t(3,n) for certain Ramsey numbers.

Determined v(3,7)=18 and v(3,8)=22.

Solved four open cases from previous studies.

Abstract

The upper (mixed) domination Ramsey number $u(m, n)$($v(m,n)$) is the smallest integer $p$ such that every $2$-coloring of the edges of $K_p$ with color red and blue, $\Gamma(B) \geq m$ or $\Gamma(R) \geq n$ ($\beta(R) \geq n$); where $B$ and $R$ is the subgraph of $K_p$ induced by blue and red edges, respectively; $\Gamma(G)$ is the maximum cardinality of a minimal dominating set of a graph $G$. First, we prove that $v(3,n)=t(3,n)$ where $t(m,n)$ is the mixed irredundant Ramsey number i.e. the smallest integer $p$ such that in every two-coloring $(R, B)$ of the edges of $K_p$, $IR(B) \geq m$ or $\beta(R) \geq n$ ($IR(G)$ is the maximum cardinality of an irredundant set of $G$). To achieve this result we use a characterization of the upper domination perfect graphs in terms of forbidden induced subgraphs. By the equality we determine two previously unknown Ramsey numbers, namely $v(3,7)=18$ and $v(3,8) = 22$. In addition, we solve other four remaining open cases from Burger's {\it et. al.} article, which listed all nonclassical Ramsey numbers. We find that $u(3,7)=w(7,3)=18$, $u(3,8) = w(8,3) = 21$, where $w(m,n)$ is the irredundant-domination Ramsey number introduced by Burger and Van Vuuren in 2011.