Group Colorings and DP-Colorings of Multigraphs Using Edge-Disjoint Decompositions

In (J Graph Theory 4:241–242, 1980), Burr proved that χ ( G ) ≤ m 1 m 2 … m k if and only if G is the edge-disjoint union of k graphs G 1 , G 2 , … , G k such that χ ( G i ) ≤ m i for 1 ≤ i ≤ k . This result established the practice of describing the chromatic number of a graph G which is the edge-disjoint union of k subgraphs G 1 , G 2 , … , G k in terms of the chromatic numbers of these subgraphs, and more specific results and conjectures followed. We investigate possible extensions of this theorem of Burr to group coloring and DP-coloring of multigraphs, as well as extensions of another vertex coloring theorem involving arboricity. In particular, we determine the DP-chromatic number of all Halin graphs. In (J Graph Theory 50:123–129, 2005), it is conjectured that for any graph G , the list chromatic number is not higher than the group chromatic number of G . As related results, we show that the group list chromatic number of all multigraphs is at most the DP-chromatic number, and present an example G for which the group chromatic number of G is less than the DP-chromatic number of G .