Coloring squares of planar graphs with girth six

Wang and Lih conjectured that for every g ≥ 5 , there exists a number M ( g ) such that the square of a planar graph G of girth at least g and maximum degree Δ ≥ M ( g ) is ( Δ + 1 ) -colorable. The conjecture is known to be true for g ≥ 7 but false for g ∈ { 5 , 6 } . We show that the conjecture for g = 6 is off by just one, i.e., the square of a planar graph G of girth at least six and sufficiently large maximum degree is ( Δ + 2 ) -colorable.