809 research outputs found
Cyclic Coloring of Plane Graphs with Maximum Face Size 16 and 17
Plummer and Toft conjectured in 1987 that the vertices of every 3-connected
plane graph with maximum face size D can be colored using at most D+2 colors in
such a way that no face is incident with two vertices of the same color. The
conjecture has been proven for D=3, D=4 and D>=18. We prove the conjecture for
D=16 and D=17
Chromatic numbers of exact distance graphs
For any graph G = (V;E) and positive integer p, the exact distance-p graph G[\p] is the graph with vertex set V , which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[\p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[\p] is bounded by the weak (2
Subchromatic numbers of powers of graphs with excluded minors
A -subcolouring of a graph is a function
such that the set of vertices coloured induce a disjoint union of cliques.
The subchromatic number, , is the minimum such that
admits a -subcolouring. Ne\v{s}et\v{r}il, Ossona de Mendez, Pilipczuk,
and Zhu (2020), recently raised the problem of finding tight upper bounds for
when is planar. We show that
when is planar, improving their bound of
135. We give even better bounds when the planar graph has larger girth.
Moreover, we show that , improving the
previous bound of 364. For these we adapt some recent techniques of Almulhim
and Kierstead (2022), while also extending the decompositions of triangulated
planar graphs of Van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich and
Siebertz (2017), to planar graphs of arbitrary girth. Note that these
decompositions are the precursors of the graph product structure theorem of
planar graphs.
We give improved bounds for for all , whenever
has bounded treewidth, bounded simple treewidth, bounded genus, or excludes
a clique or biclique as a minor. For this we introduce a family of parameters
which form a gradation between the strong and the weak colouring numbers. We
give upper bounds for these parameters for graphs coming from such classes.
Finally, we give a 2-approximation algorithm for the subchromatic number of
graphs coming from any fixed class with bounded layered cliquewidth. In
particular, this implies a 2-approximation algorithm for the subchromatic
number of powers of graphs coming from any fixed class with bounded
layered treewidth (such as the class of planar graphs). This algorithm works
even if the power and the graph is unknown.Comment: 21 pages, 2 figure
Third case of the Cyclic Coloring Conjecture
The Cyclic Coloring Conjecture asserts that the vertices of every plane graph
with maximum face size D can be colored using at most 3D/2 colors in such a way
that no face is incident with two vertices of the same color. The Cyclic
Coloring Conjecture has been proven only for two values of D: the case D=3 is
equivalent to the Four Color Theorem and the case D=4 is equivalent to
Borodin's Six Color Theorem, which says that every graph that can be drawn in
the plane with each edge crossed by at most one other edge is 6-colorable. We
prove the case D=6 of the conjecture
A Unified Approach to Distance-Two Colouring of Graphs on Surfaces
In this paper we introduce the notion of -colouring of a graph :
For given subsets of neighbours of , for every , this
is a proper colouring of the vertices of such that, in addition, vertices
that appear together in some receive different colours. This
concept generalises the notion of colouring the square of graphs and of cyclic
colouring of graphs embedded in a surface. We prove a general result for graphs
embeddable in a fixed surface, which implies asymptotic versions of Wegner's
and Borodin's Conjecture on the planar version of these two colourings. Using a
recent approach of Havet et al., we reduce the problem to edge-colouring of
multigraphs, and then use Kahn's result that the list chromatic index is close
to the fractional chromatic index.
Our results are based on a strong structural lemma for graphs embeddable in a
fixed surface, which also implies that the size of a clique in the square of a
graph of maximum degree embeddable in some fixed surface is at most
plus a constant.Comment: 36 page
- …