We initiate the study of applying the Combinatorial Nullstellensatz to the DP-coloring of graphs even though, as is well-known, the Alon-Tarsi theorem does not apply to DP-coloring. We define the notion of good covers of prime order which allows us to apply the Combinatorial Nullstellensatz to DP-coloring. We apply these tools to DP-coloring of the cones of certain bipartite graphs and uniquely 3-colorable graphs. We also extend a result of Akbari, Mirrokni, and Sadjad (2006) on unique list colorability to the context of DP-coloring. We establish a sufficient algebraic condition for a graph $G$ to satisfy $\chi_{DP}(G) \leq 3$, and we completely determine the DP-chromatic number of squares of all cycles.

中文翻译：

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图的组合 Nullstellensatz 和 DP 着色

我们开始研究将组合 Nullstellensatz 应用于图形的 DP 着色，尽管众所周知，Alon-Tarsi 定理不适用于 DP 着色。我们定义了质数阶的良好覆盖的概念，它允许我们将组合 Nullstellensatz 应用于 DP 着色。我们将这些工具应用于某些二部图和独特的 3 色图的锥体的 DP 着色。我们还将 Akbari、Mirrokni 和 Sadjad（2006）关于独特列表可着色性的结果扩展到 DP 着色的上下文。我们为图 $G$ 建立了一个充分的代数条件以满足 $\chi_{DP}(G) \leq 3$，并且我们完全确定了所有循环的 DP 色平方数。