140 research outputs found
k-forested choosability of graphs with bounded maximum average degree
A proper vertex coloring of a simple graph is -forested if the graph
induced by the vertices of any two color classes is a forest with maximum
degree less than . A graph is -forested -choosable if for a given list
of colors associated with each vertex , there exists a -forested
coloring of such that each vertex receives a color from its own list. In
this paper, we prove that the -forested choosability of a graph with maximum
degree is at most ,
or if its
maximum average degree is less than 12/5, $8/3 or 3, respectively.Comment: Please cite this paper in press as X. Zhang, G. Liu, J.-L. Wu,
k-forested choosability of graphs with bounded maximum average degree,
Bulletin of the Iranian Mathematical Society, to appea
Some results on (a:b)-choosability
A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing
that if a graph is -choosable, and , then is not
necessarily -choosable. Applying probabilistic methods, an upper bound
for the choice number of a graph is given. We also prove that a
directed graph with maximum outdegree and no odd directed cycle is
-choosable for every . Other results presented in this
article are related to the strong choice number of graphs (a generalization of
the strong chromatic number). We conclude with complexity analysis of some
decision problems related to graph choosability
Majority choosability of digraphs
A \emph{majority coloring} of a digraph is a coloring of its vertices such
that for each vertex , at most half of the out-neighbors of has the same
color as . A digraph is \emph{majority -choosable} if for any
assignment of lists of colors of size to the vertices there is a majority
coloring of from these lists. We prove that every digraph is majority
-choosable. This gives a positive answer to a question posed recently by
Kreutzer, Oum, Seymour, van der Zypen, and Wood in \cite{Kreutzer}. We obtain
this result as a consequence of a more general theorem, in which majority
condition is profitably extended. For instance, the theorem implies also that
every digraph has a coloring from arbitrary lists of size three, in which at
most of the out-neighbors of any vertex share its color. This solves
another problem posed in \cite{Kreutzer}, and supports an intriguing conjecture
stating that every digraph is majority -colorable
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
Acyclic 4-choosability of planar graphs without 4-cycles
summary:A proper vertex coloring of a graph is acyclic if there is no bicolored cycle in . In other words, each cycle of must be colored with at least three colors. Given a list assignment , if there exists an acyclic coloring of such that for all , then we say that is acyclically -colorable. If is acyclically -colorable for any list assignment with for all , then is acyclically -choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without 4-cycles is acyclically 4-choosable. However, this has been as yet verified only for some restricted classes of planar graphs. In this paper, we prove that every planar graph with neither 4-cycles nor intersecting -cycles for each is acyclically 4-choosable
- …