1,504 research outputs found
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Defective and Clustered Choosability of Sparse Graphs
An (improper) graph colouring has "defect" if each monochromatic subgraph
has maximum degree at most , and has "clustering" if each monochromatic
component has at most vertices. This paper studies defective and clustered
list-colourings for graphs with given maximum average degree. We prove that
every graph with maximum average degree less than is
-choosable with defect . This improves upon a similar result by Havet and
Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with
maximum average degree , no bound on the number of colours
was previously known. The above result with solves this problem. It
implies that every graph with maximum average degree is
-choosable with clustering 2. This extends a
result of Kopreski and Yu [Discrete Math., 2017] to the setting of
choosability. We then prove two results about clustered choosability that
explore the trade-off between the number of colours and the clustering. In
particular, we prove that every graph with maximum average degree is
-choosable with clustering , and is
-choosable with clustering . As an
example, the later result implies that every biplanar graph is 8-choosable with
bounded clustering. This is the best known result for the clustered version of
the earth-moon problem. The results extend to the setting where we only
consider the maximum average degree of subgraphs with at least some number of
vertices. Several applications are presented
Colouring exact distance graphs of chordal graphs
For a graph and positive integer , the exact distance- graph
is the graph with vertex set and with an edge between
vertices and if and only if and have distance . Recently,
there has been an effort to obtain bounds on the chromatic number
of exact distance- graphs for from certain
classes of graphs. In particular, if a graph has tree-width , it has
been shown that for odd ,
and for even . We
show that if is chordal and has tree-width , then for odd , and for even .
If we could show that for every graph of tree-width there is a
chordal graph of tree-width which contains as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width . While we cannot do this, we show that for every graph
of genus there is a graph which is a triangulation of genus and
contains as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which
arise from reviewers' comment
Realizing the chromatic numbers and orders of spinal quadrangulations of surfaces
A method is suggested for construction of quadrangulations of the closed
orientable surface with given genus g and either (1) with given chromatic
number or (2) with given order allowed by the genus g. In particular, N.
Hartsfield and G. Ringel's results [Minimal quadrangulations of orientable
surfaces, J. Combin. Theory, Series B 46 (1989) 84-95] are generalized by way
of generating new minimal quadrangulations of infinitely many other genera.Comment: 6 pages. This version is only slightly different from the original
version submitted on 8 Jul 2012: the author's affiliation has been changed
and the presentation has been slightly improve
Universal targets for homomorphisms of edge-colored graphs
A -edge-colored graph is a finite, simple graph with edges labeled by
numbers . A function from the vertex set of one -edge-colored
graph to another is a homomorphism if the endpoints of any edge are mapped to
two different vertices connected by an edge of the same color. Given a class
of graphs, a -edge-colored graph (not necessarily
with the underlying graph in ) is -universal for
when any -edge-colored graph with the underlying graph in
admits a homomorphism to . We characterize graph classes that admit
-universal graphs. For such classes, we establish asymptotically almost
tight bounds on the size of the smallest universal graph.
For a nonempty graph , the density of is the maximum ratio of the
number of edges to the number of vertices ranging over all nonempty subgraphs
of . For a nonempty class of graphs, denotes
the density of , that is the supremum of densities of graphs in
.
The main results are the following. The class admits
-universal graphs for if and only if there is an absolute constant
that bounds the acyclic chromatic number of any graph in . For any
such class, there exists a constant , such that for any , the size
of the smallest -universal graph is between and
.
A connection between the acyclic coloring and the existence of universal
graphs was first observed by Alon and Marshall (Journal of Algebraic
Combinatorics, 8(1):5-13, 1998). One of their results is that for planar
graphs, the size of the smallest -universal graph is between and
. Our results yield that there exists a constant such that for all
, this size is bounded from above by
Boxicity of graphs on surfaces
The boxicity of a graph is the least integer for which there
exist interval graphs , , such that . Scheinerman proved in 1984 that outerplanar graphs have boxicity
at most two and Thomassen proved in 1986 that planar graphs have boxicity at
most three. In this note we prove that the boxicity of toroidal graphs is at
most 7, and that the boxicity of graphs embeddable in a surface of
genus is at most . This result yields improved bounds on the
dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure
- …