2-distance coloring of sparse graphs

A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most 2 receive distinct colors. We prove that every graph with maximum degree Δ at least 4 and maximum average degree less that 7 3 admits a 2-distance ( Δ + 1 ) -coloring. This result is tight. This improves previous known results of Dolama and Sopena.