122 research outputs found
Improper choosability and Property B
A fundamental connection between list vertex colourings of graphs and
Property B (also known as hypergraph 2-colourability) was already known to
Erd\H{o}s, Rubin and Taylor. In this article, we draw similar connections for
improper list colourings. This extends results of Kostochka, Alon, and Kr\'al'
and Sgall for, respectively, multipartite graphs, graphs of large minimum
degree, and list assignments with bounded list union.Comment: 12 page
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
The inapproximability for the (0,1)-additive number
An
{\it additive labeling} of a graph is a function , such that for every two adjacent vertices and of , ( means that is joined to ). The {\it additive number} of ,
denoted by , is the minimum number such that has a additive
labeling . The {\it additive
choosability} of a graph , denoted by , is the smallest
number such that has an additive labeling for any assignment of lists
of size to the vertices of , such that the label of each vertex belongs
to its own list.
Seamone (2012) \cite{a80} conjectured that for every graph , . We give a negative answer to this conjecture and we show that
for every there is a graph such that .
A {\it -additive labeling} of a graph is a function , such that for every two adjacent vertices and
of , .
A graph may lack any -additive labeling. We show that it is -complete to decide whether a -additive labeling exists for
some families of graphs such as perfect graphs and planar triangle-free graphs.
For a graph with some -additive labelings, the -additive
number of is defined as where is the set of -additive labelings of .
We prove that given a planar graph that admits a -additive labeling, for
all , approximating the -additive number within is -hard.Comment: 14 pages, 3 figures, Discrete Mathematics & Theoretical Computer
Scienc
Few Long Lists for Edge Choosability of Planar Cubic Graphs
It is known that every loopless cubic graph is 4-edge choosable. We prove the
following strengthened result.
Let G be a planar cubic graph having b cut-edges. There exists a set F of at
most 5b/2 edges of G with the following property. For any function L which
assigns to each edge of F a set of 4 colours and which assigns to each edge in
E(G)-F a set of 3 colours, the graph G has a proper edge colouring where the
colour of each edge e belongs to L(e).Comment: 14 pages, 1 figur
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