Chromatic number of graphs and edge Folkman numbers

TL;DR

This paper establishes a lower bound on the number of vertices in a graph based on its chromatic number, identifies graphs where this bound is tight, and applies these findings to edge Folkman numbers.

Contribution

It introduces a new lower bound for graph vertices related to chromatic number and explores graphs where this bound is exact, with applications to Folkman numbers.

Findings

Derived a lower bound for vertices using chromatic number

Characterized graphs where the bound is exact

Applied results to edge Folkman numbers

Abstract

In the paper we give a lower bound for the number of vertices of a given graph using its chromatic number. We find the graphs for which this bound is exact. The results are applied in the theory of Foklman numbers.