A bound on the chromatic number of the square of a planar graph

Wegner conjectured that the chromatic number of the square of any planar graph G with maximum degree Δ ⩾ 8 is bounded by χ ( G 2 ) ⩽ ⌊ 3 2 Δ ⌋ + 1 . We prove the bound χ ( G 2 ) ⩽ ⌈ 5 3 Δ ⌉ + 78 . This is asymptotically an improvement on the previously best-known bound. For large values of Δ we give the bound of χ ( G 2 ) ⩽ ⌈ 5 3 Δ ⌉ + 25 . We generalize this result to L ( p , q ) -labeling of planar graphs, by showing that λ q p ( G ) ⩽ q ⌈ 5 3 Δ ⌉ + 18 p + 77 q - 18 . For each of the results, the proof provides a quadratic time algorithm.