List-Coloring Squares of Sparse Subcubic Graphs

The problem of colouring the square of a graph naturally arises in connection with the distance labelings, which have been studied intensively. We consider this problem for sparse subcubic graphs and show that the choosability χ`(G 2) of the square of a subcubic graph G of maximum average degree d is at most four if d < 24/11 and G does not contain a 5-cycle, χ`(G 2) is at most five if d < 7/3 and at most six if d < 5/2. Wegner’s conjecture claims that the chromatic number of the square of a subcubic planar graph is at most seven. Our result implies that χ`(G 2) is at most four if g ≥ 24, it is at most 5 if g ≥ 14, and it is at most 6 if g ≥ 10. For lower bounds, we find a planar subcubic graph G1 of girth 9 such that χ(G1) = 5 and a planar subcubic graph G2 of girth five such that χ(G2) = 6. As a consequence, we show that the problem of 4-colouring of the square of a subcubic planar graph of girth g = 9 is NP-complete. We conclude the paper by posing few conjectures.