Abstract For a graph G with at least one vertex with independent neighborhood, an adynamic coloring of G is a proper vertex coloring of G such that there exists at least one vertex of degree at least 2 whose all neighbors have the same color. We explore basic properties of adynamic colorings and their relations to proper and dynamic colorings. We also establish a number of results for planar graphs; in particular, we extend the Four Color Theorem and Grotzsch’s Theorem to adynamic coloring. Finally, we prove that triangle-free graphs with maximum degree 3 are adynamically 3-colorable, which is surprisingly not true for graphs of higher maximum degrees.

中文翻译：

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图形的动态着色

摘要 对于具有至少一个具有独立邻域的顶点的图 G，G 的动态着色是 G 的适当顶点着色，使得至少存在一个度数至少为 2 的顶点的所有邻居具有相同的颜色。我们探索非动态着色的基本特性及其与适当和动态着色的关系。我们还为平面图建立了许多结果；特别是，我们将四色定理和 Grotzsch 定理扩展到动态着色。最后，我们证明了最大度数为 3 的无三角形图是动态 3 可着色的，这对于最大度数更高的图令人惊讶地不是真的。