Coloring squares of planar graphs with small maximum degree

For a graph G, by 2 (G) we denote the minimum integer k, such that there is a k-coloring of the vertices of G in which vertices at distance at most 2 receive distinct colors. Equivalently, 2 (G) is the chromatic number of the square of G. In 1977 Wegner conjectured that if G is planar and has maximum degree , then 2 (G) 7 if 3, 2 (G) + 5 if 4 7, and 3/2 + 1 if 8. Despite extensive work, the known upper bounds are quite far from the conjectured ones, especially for small values of . In this work we show that for every planar graph G with maximum degree it holds that 2 (G) 3 + 4. This result provides the best known upper bound for 6 14.