5 research outputs found
Clustered Colouring in Minor-Closed Classes
The "clustered chromatic number" of a class of graphs is the minimum integer
such that for some integer every graph in the class is -colourable
with monochromatic components of size at most . We prove that for every
graph , the clustered chromatic number of the class of -minor-free graphs
is tied to the tree-depth of . In particular, if is connected with
tree-depth then every -minor-free graph is -colourable with
monochromatic components of size at most . This provides the first
evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about
defective colouring of -minor-free graphs. If then we prove that 4
colours suffice, which is best possible. We also determine those minor-closed
graph classes with clustered chromatic number 2. Finally, we develop a
conjecture for the clustered chromatic number of an arbitrary minor-closed
class
Clustered 3-Colouring Graphs of Bounded Degree
A (not necessarily proper) vertex colouring of a graph has "clustering"
if every monochromatic component has at most vertices. We prove that planar
graphs with maximum degree are 3-colourable with clustering
. The previous best bound was . 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 that exclude a fixed minor are 3-colourable
with clustering . The best previous bound for this result was
exponential in .Comment: arXiv admin note: text overlap with arXiv:1904.0479
Clustered Graph Coloring and Layered Treewidth
A graph coloring has bounded clustering if each monochromatic component has
bounded size. This paper studies clustered coloring, where the number of colors
depends on an excluded complete bipartite subgraph. This is a much weaker
assumption than previous works, where typically the number of colors depends on
an excluded minor. This paper focuses on graph classes with bounded layered
treewidth, which include planar graphs, graphs of bounded Euler genus, graphs
embeddable on a fixed surface with a bounded number of crossings per edge,
amongst other examples. Our main theorem says that for fixed integers ,
every graph with layered treewidth at most and with no subgraph
is -colorable with bounded clustering. In the case, which
corresponds to graphs of bounded maximum degree, we obtain polynomial bounds on
the clustering. This greatly improves a corresponding result of Esperet and
Joret for graphs of bounded genus. The case implies that every graph with
a drawing on a fixed surface with a bounded number of crossings per edge is
5-colorable with bounded clustering. Our main theorem is also a critical
component in two companion papers that study clustered coloring of graphs with
no -subgraph and excluding a fixed minor, odd minor or topological
minor