55 research outputs found
Beyond Ohba's Conjecture: A bound on the choice number of -chromatic graphs with vertices
Let denote the choice number of a graph (also called "list
chromatic number" or "choosability" of ). Noel, Reed, and Wu proved the
conjecture of Ohba that when . We
extend this to a general upper bound: . Our result is sharp for
using Ohba's examples, and it improves the best-known
upper bound for .Comment: 14 page
Choosability of a weighted path and free-choosability of a cycle
A graph with a list of colors and weight for each vertex
is -colorable if one can choose a subset of colors from
for each vertex , such that adjacent vertices receive disjoint color
sets. In this paper, we give necessary and sufficient conditions for a weighted
path to be -colorable for some list assignments . Furthermore, we
solve the problem of the free-choosability of a cycle.Comment: 9 page
Defective and Clustered Choosability of Sparse Graphs
An (improper) graph colouring has "defect" if each monochromatic subgraph
has maximum degree at most , and has "clustering" if each monochromatic
component has at most vertices. This paper studies defective and clustered
list-colourings for graphs with given maximum average degree. We prove that
every graph with maximum average degree less than is
-choosable with defect . This improves upon a similar result by Havet and
Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with
maximum average degree , no bound on the number of colours
was previously known. The above result with solves this problem. It
implies that every graph with maximum average degree is
-choosable with clustering 2. This extends a
result of Kopreski and Yu [Discrete Math., 2017] to the setting of
choosability. We then prove two results about clustered choosability that
explore the trade-off between the number of colours and the clustering. In
particular, we prove that every graph with maximum average degree is
-choosable with clustering , and is
-choosable with clustering . As an
example, the later result implies that every biplanar graph is 8-choosable with
bounded clustering. This is the best known result for the clustered version of
the earth-moon problem. The results extend to the setting where we only
consider the maximum average degree of subgraphs with at least some number of
vertices. Several applications are presented
Every triangle-free induced subgraph of the triangular lattice is -choosable
International audienceA graph is -choosable if for any color list of size associated with each vertex, one can choose a subset of colors such that adjacent vertices are colored with disjoint color sets. This paper proves that for any integer , every finite triangle-free induced subgraph of the triangular lattice is -choosable
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