22,405 research outputs found
Weak degeneracy of graphs
Motivated by the study of greedy algorithms for graph coloring, we introduce
a new graph parameter, which we call weak degeneracy. By definition, every
-degenerate graph is also weakly -degenerate. On the other hand, if
is weakly -degenerate, then (and, moreover, the same
bound holds for the list-chromatic and even the DP-chromatic number of ). It
turns out that several upper bounds in graph coloring theory can be phrased in
terms of weak degeneracy. For example, we show that planar graphs are weakly
-degenerate, which implies Thomassen's famous theorem that planar graphs are
-list-colorable. We also prove a version of Brooks's theorem for weak
degeneracy: a connected graph of maximum degree is weakly
-degenerate unless . (By contrast, all -regular
graphs have degeneracy .) We actually prove an even stronger result, namely
that for every , there is such that if is a graph
of weak degeneracy at least , then either contains a -clique or
the maximum average degree of is at least . Finally, we show
that graphs of maximum degree and either of girth at least or of
bounded chromatic number are weakly -degenerate, which
is best possible up to the value of the implied constant.Comment: 21 p
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
List (d,1)-total labelling of graphs embedded in surfaces
The (d,1)-total labelling of graphs was introduced by Havet and Yu. In this
paper, we consider the list version of (d,1)-total labelling of graphs. Let G
be a graph embedded in a surface with Euler characteristic whose
maximum degree is sufficiently large. We prove that the (d,1)-total
choosability of is at most .Comment: 6 page
A Linear-Time Algorithm for Finding Induced Planar Subgraphs
In this paper we study the problem of efficiently and effectively extracting induced planar subgraphs. Edwards and Farr proposed an algorithm with O(mn) time complexity to find an induced planar subgraph of at least 3n/(d+1) vertices in a graph of maximum degree d. They also proposed an alternative algorithm with O(mn) time complexity to find an induced planar subgraph graph of at least 3n/(bar{d}+1) vertices, where bar{d} is the average degree of the graph. These two methods appear to be best known when d and bar{d} are small. Unfortunately, they sacrifice accuracy for lower time complexity by using indirect indicators of planarity. A limitation of those approaches is that the algorithms do not implicitly test for planarity, and the additional costs of this test can be significant in large graphs. In contrast, we propose a linear-time algorithm that finds an induced planar subgraph of n-nu vertices in a graph of n vertices, where nu denotes the total number of vertices shared by the detected Kuratowski subdivisions. An added benefit of our approach is that we are able to detect when a graph is planar, and terminate the reduction. The resulting planar subgraphs also do not have any rigid constraints on the maximum degree of the induced subgraph. The experiment results show that our method achieves better performance than current methods on graphs with small skewness
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