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In the present paper we extend the following three coloring concepts for the class of finite undirected graphs having multiple edges but no loops. First of all, the generalized coloring concept, in which the same colored vertices of a graph induce a subgraph satisfying a prescribed graph property. Secondly, the concept of variable degeneracy, which was introduced by Borodin, Kostochka and Toft in 2000; this makes it possible to give a common generalization of the point partition number and the list chromatic number. Finally, the DP-coloring concept as introduced by Ďvorák and Postle in 2018, where a list assignment of a graph is replaced by a cover. Combining these three coloring concepts leads to generalizations of various classical coloring results, including the theorems of Brooks, of Gallai, and of Erdős, Rubin and Taylor. Our main result is a DP-version of a theorem about partitions of graphs into a fixed number of induced subgraphs with bounded variable degeneracy due to Borodin, Kostochka, and Toft.
Brooks' Theorem
(2024)
In the present paper we extend the following three coloring concepts for the class of finite undirected graphs having multiple edges but no loops. First of all, the generalized coloring concept, in which the same colored vertices of a graph induce a subgraph satisfying a prescribed graph property. Secondly, the concept of variable degeneracy, which was introduced by Borodin, Kostochka and Toft in 2000; this makes it possible to give a common generalization of the point partition number and the list chromatic number. Finally, the DP-coloring concept as introduced by Ďvorák and Postle in 2018, where a list assignment of a graph is replaced by a cover. Combining these three coloring concepts leads to generalizations of various classical coloring results, including the theorems of Brooks, of Gallai, and of Erdős, Rubin and Taylor. Our main result is a DP-version of a theorem about partitions of graphs into a fixed number of induced subgraphs with bounded variable degeneracy due to Borodin, Kostochka, and Toft.
The paper deals with partitions of hypergraphs into induced subhypergraphs satisfying constraints on their degeneracy. Our hypergraphs may have multiple edges, but no loops. Given a hypergraph and a sequence of vertex functions such that for all , we want to find a sequence of vertex disjoint induced subhypergraphs containing all vertices of such that each hypergraph is strictly ‐degenerate, that is, for every nonempty subhypergraph there is a vertex such that . Our main result in this paper says that such a sequence of hypergraphs exists if and only if is not a so‐called hard pair. Hard pairs form a recursively defined family of configurations, obtained from three basic types of configurations by the operation of merging a vertex. Our main result has several interesting applications related to generalized hypergraph coloring problems.
On DP‐coloring of digraphs
(2019)
DP‐coloring is a relatively new coloring concept by Dvořák and Postle and was introduced as an extension of list‐colorings of (undirected) graphs. It transforms the problem of finding a list‐coloring of a given graph with a list‐assignment to finding an independent transversal in an auxiliary graph with vertex set . In this paper, we extend the definition of DP‐colorings to digraphs using the approach from Neumann‐Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks’ type theorem regarding the DP‐chromatic number, which extends various results on the (list‐)chromatic number of digraphs.
