2-distance 4-colorability of Planar Subcubic Graphs with Girth at Least 22

The trivial lower bound for the 2-distance chromatic number χ2(G) of any graph G with maximum degree ∆ is ∆+1. It is known that χ2 = ∆+1 if the girth g of G is at least 7 and ∆ is large enough. There are graphs with arbitrarily large ∆ and g ≤ 6 having χ2(G) ≥ ∆ + 2. We prove the 2-distance 4-colorability of planar subcubic graphs with g ≥ 22.

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