6 research outputs found
Graph product structure for non-minor-closed classes
Dujmovi\'c et al. (FOCS 2019) recently proved that every planar graph is a
subgraph of the strong product of a graph of bounded treewidth and a path.
Analogous results were obtained for graphs of bounded Euler genus or
apex-minor-free graphs. These tools have been used to solve longstanding
problems on queue layouts, non-repetitive colouring, -centered colouring,
and adjacency labelling. This paper proves analogous product structure theorems
for various non-minor-closed classes. One noteable example is -planar graphs
(those with a drawing in the plane in which each edge is involved in at most
crossings). We prove that every -planar graph is a subgraph of the
strong product of a graph of treewidth and a path. This is the first
result of this type for a non-minor-closed class of graphs. It implies, amongst
other results, that -planar graphs have non-repetitive chromatic number
upper-bounded by a function of . All these results generalise for drawings
of graphs on arbitrary surfaces. In fact, we work in a much more general
setting based on so-called shortcut systems that are of independent interest.
This leads to analogous results for map graphs, string graphs, graph powers,
and nearest neighbour graphs.Comment: v2 Cosmetic improvements and a corrected bound for
(layered-)(tree)width in Theorems 2, 9, 11, and Corollaries 1, 3, 4, 6, 12.
v3 Complete restructur
On the Strong Parity Chromatic Number
International audienceA vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. [Discrete Math. 311 (2011) 512-520] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of plane graphs
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