Colourings with Bounded Monochromatic Components in Graphs of Given   Circumference
  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. 
Bojan Mohar and Bruce Reed and David R. Wood
math.CO