5 research outputs found
Defective 3-Paintability of Planar Graphs
A -defective -painting game on a graph is played by two players:
Lister and Painter. Initially, each vertex is uncolored and has tokens. In
each round, Lister marks a chosen set of uncolored vertices and removes one
token from each marked vertex. In response, Painter colors vertices in a subset
of which induce a subgraph of maximum degree at most . Lister
wins the game if at the end of some round there is an uncolored vertex that has
no more tokens left. Otherwise, all vertices eventually get colored and Painter
wins the game. We say that is -defective -paintable if Painter has a
winning strategy in this game. In this paper we show that every planar graph is
3-defective 3-paintable and give a construction of a planar graph that is not
2-defective 3-paintable.Comment: 21 pages, 11 figure
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