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Further Extensions of the Grötzsch Theorem

The Grötzsch Theorem states that every triangle-free planar graph admits a proper $3$-coloring. Among many of its generalizations, the one of Grünbaum and Aksenov, giving $3$-colorability of planar graphs with at most three triangles, is perhaps the most known. A lot of attention was also given to extending $3$-colorings of subgraphs to the whole graph. In this paper, we consider $3$-colorings of planar graphs with at most one triangle. Particularly, we show that precoloring of any two non-adjacent vertices and precoloring of a face of length at most $4$ can be extended to a $3$-coloring of the graph. Additionally, we show that for every vertex of degree at most $3$, a precoloring of its neighborhood with the same color extends to a $3$-coloring of the graph. The latter result implies an affirmative answer to a conjecture on adynamic coloring. All the presented results are tight.