Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Further Extensions of the Gr\"{o}tzsch Theorem   The Gr\"{o}tzsch Theorem states that every triangle-free planar graph admits
a proper $3$-coloring. Among many of its generalizations, the one of
Gr\"{u}nbaum 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.

26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
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The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg