Number of vertices in graphs with locally small chromatic number and large chromatic number

TL;DR

This paper investigates the relationship between local and global chromatic numbers in graphs, establishing bounds on the number of vertices needed for graphs with small local chromatic number but large overall chromatic number.

Contribution

It provides a new bound on the minimum number of vertices in graphs with small local chromatic number and large global chromatic number, extending understanding of graph coloring properties.

Findings

Graphs with approximately ((n+rc)/(c+rc))^{r+1} vertices can have a large chromatic number despite small local chromatic numbers.

The paper establishes an upper bound on the chromatic number based on the number of vertices and local properties.

It advances theoretical understanding of how local constraints influence global graph coloring complexity.

Abstract

We discuss the minimal number of vertices in a graph with a large chromatic number such that each ball of a fixed radius in it has a small chromatic number. It is shown that for every graph $G$ on $\sim((n+rc)/(c+rc))^{r+1}$ vertices such that each ball of radius $r$ is properly $c$-colorable, we have $\chi(G)\leq n$.