Colourings with Bounded Monochromatic Components in Graphs of Given Circumference

TL;DR

This paper proves that graphs with limited circumference can be coloured with a logarithmic number of colours to ensure small monochromatic components, establishing optimal bounds.

Contribution

It introduces a tight bound on the number of colours needed for such graphs, linking circumference to monochromatic component size.

Findings

Graphs with circumference at most k are O(log k)-colourable.

Monochromatic components have size at most O(k).

The O(log k) bound is proven to be optimal.

Abstract

We prove that every graph with circumference at most $k$ is $O(\log k)$-colourable such that every monochromatic component has size at most $O(k)$. The $O(\log k)$ bound on the number of colours is best possible, even in the setting of colourings with bounded monochromatic degree.