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.

在 (J Graph Theory 4:241–242, 1980) 中，Burr 证明了\(\chi (G)\le m_1m_2 \ldots m_k\)当且仅当G是k 个图的边不相交并集\(G_1, G_2,\ldots ,G_k\)使得\(\chi (G_i)\le m_i\)为\(1\le i\le k\)。该结果建立了描述图G的色数的实践，该图G是k个子图的边不相交并集\(G_1,G_2,\ldots ,G_k\)就这些子图的色数而言，更具体的结果和猜想随之而来。我们研究了 Burr 定理对多重图的分组着色和 DP 着色的可能扩展，以及另一个涉及树状的顶点着色定理的扩展。特别是，我们确定了所有 Halin 图的 DP 色数。在 (J Graph Theory 50:123–129, 2005) 中，推测对于任何图G，表色数不高于G的群色数。作为相关的结果，我们表明，所有多重图的组列表色数至多为DP-色数，并呈现一个例子ģ为其组成的组色数ģ小于G的 DP 色数。