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 Grötzsch’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.
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