Geodesic
List 2-distance $(\Delta+1)$-coloring of planar graphs with girth at least~7
Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 5, pp. 22-36

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A trivial lower bound for the 2-distance chromatic number $\chi_2(G)$ of every graph $G$ with maximum degree $\Delta$ is $\Delta+1$. There are graphs with arbitrarily large $\Delta$ and girth $g\le6$ having $\chi_2(G)\ge\Delta+2$. In the paper are improved previously known restrictions on $\Delta$ and $g$ under which every planar graph $G$ has $\chi_2(G)=\Delta+1$. Ill. 2, bibliogr. 24.
Keywords: planar graph, 2-distance coloring, list coloring. 
@article{DA_2010_17_5_a2,
     author = {A. O. Ivanova},
     title = {List 2-distance $(\Delta+1)$-coloring of planar graphs with girth at least~7},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {22--36},
     publisher = {mathdoc},
     volume = {17},
     number = {5},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2010_17_5_a2/}
}
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A. O. Ivanova. List 2-distance $(\Delta+1)$-coloring of planar graphs with girth at least~7. Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 5, pp. 22-36. http://geodesic.mathdoc.fr/item/DA_2010_17_5_a2/