Coloring squares of planar graphs with small maximum degree

Abstract

For a graph G, by χ 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, χ 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 χ 2 (G) ≤ 7 if ∆ ≤ 3, χ 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 χ 2 (G) ≤ 3∆ + 4. This result provides the best known upper bound for 6 ≤ ∆ ≤ 14.