356 research outputs found
On edge-group choosability of graphs
In this paper, we study the concept of edge-group choosability of graphs. We
say that G is edge k-group choosable if its line graph is k-group choosable. An
edge-group choosability version of Vizing conjecture is given. The evidence of
our claim are graphs with maximum degree less than 4, planar graphs with
maximum degree at least 11, planar graphs without small cycles, outerplanar
graphs and near-outerplanar graphs
Disproof of the List Hadwiger Conjecture
The List Hadwiger Conjecture asserts that every -minor-free graph is
-choosable. We disprove this conjecture by constructing a
-minor-free graph that is not -choosable for every integer
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
Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count
We show that triangle-free penny graphs have degeneracy at most two, list
coloring number (choosability) at most three, diameter , and
at most edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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