Clustered 3-colouring graphs of bounded degree

Abstract

AbstractA (not necessarily proper) vertex colouring of a graph hasclustering cif every monochromatic component has at mostcvertices. We prove that planar graphs with maximum degree$\Delta$are 3-colourable with clustering$O(\Delta^2)$. The previous best bound was$O(\Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree$\Delta$that exclude a fixed minor are 3-colourable with clustering$O(\Delta^5)$. The best previous bound for this result was exponential in$\Delta$.