On DP-coloring of digraphs

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 $G$ with a list-assignment $L$ to finding an independent transversal in an auxiliary graph with vertex set $\text{◂{}▸}\{\left(v,c\right)|\text{◂+▸}v\in V\left(G\right),\text{◂+▸}c\in L\left(v\right)\}$. 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.     

Dirac's map-color theorem for choosability

It is proved that the choice number of every graph G embedded on a surface of Euler genus ε ≥ 1 and ε ≠ 3 is at most the Heawood number and that the equality holds if and only if G contains the complete graph K_H(ε) as a subgraph. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 327–339, 1999     

A note on a Brooks' type theorem for DP-coloring

Dvořák and Postle introduced DP-coloring of simple graphs as a generalization of list-coloring. They proved a Brooks' type theorem for DP-coloring; and Bernshteyn, Kostochka, and Pron extended it to DP-coloring of multigraphs. However, detailed structure, when a multigraph does not admit DP-coloring, was not specified. In this note, we make this point clear and give the complete structure. This is also motivated by the relation to signed coloring of signed graphs.     

Graphs whose choice number is equal to their chromatic number

A graph G is k-choosable if it admits a vertex-coloring whenever the colors allowed at each vertex are restricted to a list of length k. If χ denotes the usual chromatic number of G, we are interested in which kind of G is χ-choosable. This question contains a famous conjecture, which states that every line-graph is χ-choosable. We present some other classes of graphs that are χ-choosable; all these classes are related to claw-free graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 87–97, 1998     

A characterization of (4,2)-choosable graphs

A graph $G$ is $\left(a,b\right)$-choosable if given any list assignment $L$ with $\text{◂=▸}\mid L\left(v\right)\mid =a$ for each $\text{◂+▸}v\in V\left(G\right)$ there exists a function $\phi$ such that $\text{◂⊆▸}\phi \left(v\right)\subseteq L\left(v\right)$ and $\text{◂=▸}\mid \phi \left(v\right)\mid =b$ for all $\text{◂+▸}v\in V\left(G\right)$, and whenever vertices $x$ and $y$ are adjacent $\text{◂+▸}\phi \left(x\right)\cap \phi \left(y\right)=\varnothing$. Meng, Puleo, and Zhu conjectured a characterization of (4,2)-choosable graphs. We prove their conjecture.