Coloring squares of planar graphs with no short cycles

Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the chromatic number of the square of every planar graph of girth at least g and maximum degree ∆ ≥ M(g) is ∆ + 1. We disprove the conjecture for g ∈ {5, 6} and prove the existence of the number M(g) for g ≥ 7. More generally, we show that every planar graph of girth at least 7 and maximum degree ∆ ≥ 190 + 2dp/qe has an L(p, q)-labeling of span at most 2p + q∆ − 2. For q = 1, the bound is tight for all pairs of ∆ and p. We also show that the square of every planar graph of girth at least six and sufficiently large maximum degree ∆ is (∆ + 2)-colorable.