81 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
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
Chip games and paintability
We prove that the difference between the paint number and the choice number
of a complete bipartite graph is . That answers
the question of Zhu (2009) whether this difference, for all graphs, can be
bounded by a common constant. By a classical correspondence, our result
translates to the framework of on-line coloring of uniform hypergraphs. This
way we obtain that for every on-line two coloring algorithm there exists a
k-uniform hypergraph with edges on which the strategy fails. The
results are derived through an analysis of a natural family of chip games
Sequence variations of the 1-2-3 Conjecture and irregularity strength
Karonski, Luczak, and Thomason (2004) conjectured that, for any connected
graph G on at least three vertices, there exists an edge weighting from {1,2,3}
such that adjacent vertices receive different sums of incident edge weights.
Bartnicki, Grytczuk, and Niwcyk (2009) made a stronger conjecture, that each
edge's weight may be chosen from an arbitrary list of size 3 rather than
{1,2,3}. We examine a variation of these conjectures, where each vertex is
coloured with a sequence of edge weights. Such a colouring relies on an
ordering of the graph's edges, and so two variations arise -- one where we may
choose any ordering of the edges and one where the ordering is fixed. In the
former case, we bound the list size required for any graph. In the latter, we
obtain a bound on list sizes for graphs with sufficiently large minimum degree.
We also extend our methods to a list variation of irregularity strength, where
each vertex receives a distinct sequence of edge weights.Comment: Accepted to Discrete Mathematics and Theoretical Computer Scienc
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