The rate of growth of the minimum clique size of graphs of given order and chromatic number

TL;DR

This paper investigates how the minimum clique size in graphs with a fixed order and chromatic number grows as the number of vertices increases, providing precise growth rates and improved bounds.

Contribution

It determines the growth rate of the minimum clique size for graphs with given order and chromatic number, and offers a tighter upper bound for this quantity.

Findings

Established the growth rate of Q(n, ⌈rn⌉) for fixed r.

Provided a better upper bound for Q(n, ⌈rn⌉).

Analyzed the asymptotic behavior of clique sizes in graphs.

Abstract

Let $Q(n,c)$ denote the minimum clique number over graphs with $n$ vertices and chromatic number $c$. We determine the rate of growth of of the sequence ${Q(n,\lceil rn \rceil)}_{n=1}^\infty$ for any fixed $0<r\leq 1$. We also give a better upper bound for $Q(n,\lceil rn \rceil)$.