Local And Global Colorability of Graphs

TL;DR

This paper investigates the relationship between local and global colorability in graphs, establishing bounds on the chromatic number based on local subgraph properties and analyzing random graph models.

Contribution

It provides a tight asymptotic characterization of the maximum chromatic number for graphs with locally bounded colorability, using probabilistic analysis of random graphs.

Findings

Maximum chromatic number is approximately n^{1/(r+1)} for graphs with local c-colorability.

Most small-radius neighborhoods in such graphs are 2-degenerate.

Random graphs with appropriate edge probability typically achieve these bounds.

Abstract

It is shown that for any fixed $c \geq 3$ and $r$, the maximum possible chromatic number of a graph on $n$ vertices in which every subgraph of radius at most $r$ is $c$ colorable is $\tilde{\Theta}\left(n ^ {\frac{1}{r+1}} \right)$ (that is, $n^\frac{1}{r+1}$ up to a factor poly-logarithmic in $n$). The proof is based on a careful analysis of the local and global colorability of random graphs and implies, in particular, that a random $n$-vertex graph with the right edge probability has typically a chromatic number as above and yet most balls of radius $r$ in it are $2$-degenerate.