22 research outputs found
Equitable Coloring and Equitable Choosability of Planar Graphs without chordal 4- and 6-Cycles
A graph is equitably -choosable if, for any given -uniform list
assignment , is -colorable and each color appears on at most
vertices. A graph is equitably -colorable if
the vertex set can be partitioned into independent subsets ,
, , such that for .
In this paper, we prove that if is a planar graph without chordal - and
-cycles, then is equitably -colorable and equitably -choosable
where .Comment: 21 pages,3 figure
Equitable partition of planar graphs
An equitable -partition of a graph is a collection of induced
subgraphs of such that
is a partition of and
for all . We prove that every planar graph admits an equitable
-partition into -degenerate graphs, an equitable -partition into
-degenerate graphs, and an equitable -partition into two forests and one
graph.Comment: 12 pages; revised; accepted to Discrete Mat
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