Geodesic
2-distance 4-colorability of planar subcubic graphs with girth at least 22
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 141-151

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The trivial lower bound for the 2-distance chromatic number χ₂(G) of any graph G with maximum degree Δ is Δ+1. It is known that χ₂ = Δ+1 if the girth g of G is at least 7 and Δ is large enough. There are graphs with arbitrarily large Δ and g ≤ 6 having χ₂(G) ≥ Δ+2. We prove the 2-distance 4-colorability of planar subcubic graphs with g ≥ 22.
Keywords: planar graph, subcubic graph, 2-distance coloring 
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     title = {2-distance 4-colorability of planar subcubic graphs with girth at least 22},
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Borodin, Oleg; Ivanova, Anna. 2-distance 4-colorability of planar subcubic graphs with girth at least 22. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 141-151. http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a11/