Geodesic
$2$-distance coloring of sparse planar graphs
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 1 (2004), pp. 76-90.

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Clearly, the 2-distance chromatic number $\chi_2(G)$ of any graph $G$ with maximum degree $\Delta$ is at least $\Delta+1$. We prove that if $G$ is planar and its girth is large enough (w.r.t. a fixed $\Delta$), then $\chi_2(G)=\Delta+1$. 
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O. V. Borodin; A. O. Ivanova; T. K. Neustroeva. $2$-distance coloring of sparse planar graphs. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 1 (2004), pp. 76-90. http://geodesic.mathdoc.fr/item/SEMR_2004_1_a6/

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