111 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
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
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
Facial unique-maximum colorings of plane graphs with restriction on big vertices
A facial unique-maximum coloring of a plane graph is a proper coloring of the
vertices using positive integers such that each face has a unique vertex that
receives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposed
a strengthening of the Four Color Theorem conjecturing that all plane graphs
have a facial unique-maximum coloring using four colors. This conjecture has
been disproven for general plane graphs and it was shown that five colors
suffice. In this paper we show that plane graphs, where vertices of degree at
least four induce a star forest, are facially unique-maximum 4-colorable. This
improves a previous result for subcubic plane graphs by Andova, Lidick\'y,
Lu\v{z}ar, and \v{S}krekovski (2018). We conclude the paper by proposing some
problems.Comment: 8 pages, 5 figure
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