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

Abstract

In (J Graph Theory 4:241–242, 1980), Burr proved that $\chi (G)\le m_1m_2 \ldots m_k$ if and only if G is the edge-disjoint union of k graphs $G_1,G_2,\ldots ,G_k$ such that $\chi (G_i)\le m_i$ for $1\le i\le k$. This result established the practice of describing the chromatic number of a graph G which is the edge-disjoint union of k subgraphs $G_1,G_2,\ldots ,G_k$ 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.