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Sufficient conditions for the 2-distance $$(\Delta+1)$$-colorability of planar graphs with girth 6. (Russian) Zbl 1249.05113
It is known that
(a)
$$\Delta+1$$ is a lower bound for a 2-distance chromatic number of a graph $$G$$ with maximum degree $$\Delta$$;
(b)
if $$G$$ is planar and its girth is at least 7 then, for sufficiently large $$\Delta$$, this bound is attained;
(c)
statement (b) is not true for graphs with girth 6.
For a planar graph $$G$$ with girth 6, $$\Delta\geq 179$$, and every edge incident with a vertex of degree 1 or 2, the authors prove that the 2-distance chromatic number of $$G$$ is equal to $$\Delta+1$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05C12 Distance in graphs
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