Geodesic
Sufficient conditions for planar graphs to be $2$-distance $(\Delta+1)$-colorable
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 1 (2004), pp. 129-141.

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A trivial lower bound for the $2$-distance chromatic number $\chi_2(G)$ of any graph $G$ with maximum degree $\Delta$ is $\Delta+1$. We prove that if $G$ is planar and its girth is at least $7$, then $\chi_2(G)=\Delta+1$ whenever $\Delta\ge 30$. On the other hand, we construct planar graphs with girth $5$ and $6$ that have arbitrarily large $\Delta$ and $\chi_2(G)>\Delta+1$. 
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     title = {Sufficient conditions for planar graphs to be $2$-distance $(\Delta+1)$-colorable},
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O. V. Borodin; A. N. Glebov; A. O. Ivanova; T. K. Neustroeva; V. A. Tashkinov. Sufficient conditions for planar graphs to be $2$-distance $(\Delta+1)$-colorable. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 1 (2004), pp. 129-141. http://geodesic.mathdoc.fr/item/SEMR_2004_1_a11/

[1] O.V. Borodin, Kh. Brusma, A.N. Glebov, Ya. Van den Khoivel, “Minimalnye stepeni i khromaticheskie chisla kvadratov ploskikh grafov”, Diskret. analiz i issled. operatsii. Ser. 1, 8:4 (2001), 9–33 | MR

[2] O.V. Borodin, A.O. Ivanova, T.K. Neustroeva, “2-distantsionnaya raskraska razrezhennykh ploskikh grafov”, Sibirskie elektronnye matematicheskie izvestiya, 1 (2004), 76–90

[3] T.R. Jensen, B. Toft, Graph coloring problems, John-Wiley Sons, New York, 1995 | MR | Zbl

[4] G. Wegner, Graphs with given diameter and a coloring problem, Technical Report. University of Dortmund, 1977