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

Abstract

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.

DP-3-coloring of some planar graphs

“…Dvorák and Postle [9] used this notation proved that every planar graph without cycles of lengths from 4 to 8 is 3-choosable (actually a stronger form using DP-coloring), solving a long-standing conjecture of Borodin [8]. Since then much attention was drawn on this new coloring, see for example, [2,3,4,5,6,7,12,13,14,18].…”

“…Dvořák and Postle [9] noted that Thomassen's proofs [19] for choosability can be used to show χ DP (G) ≤ 5 if G is a planar graph, and χ DP (G) ≤ 3 if G is a planar graph with no 3-cycles and 4-cycles. Some sufficient conditions were given in [12,13,18] for a planar graph to be DP-4-colorable.…”

DP-degree colorable hypergraphs

“…Of course, when dealing with DP-coloring, is it not only of interest to characterize the non DP-degree colorable graphs, but also the corresponding 'bad' covers. This was done by Kim and Ozeki [11] (see Theorem 7). The aim of this section is to give the corresponding characterizations for DP-degree-colorable hypergraphs.…”

Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable

“…Much attention was drawn on this new coloring, see for example, [2,3,4,5,6,7,18,19,20,23,22]. We are interested in DP-coloring of planar graphs.…”

“…Dvořák and Postle [13] noted that Thomassen's proofs [25] for choosability can be used to show χ DP (G) ≤ 5 if G is a planar graph, and χ DP (G) ≤ 3 if G is a planar graph with no 3-cycles and 4-cycles. Some sufficient conditions were given in [18,19,23] for a planar graph to be DP-4-colorable. Sufficient conditions for a planar graph to be DP-3-colorable are obtained in [21] and [22].…”