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(G2) ≤ ⌊3/2 Δ⌋ + 1. We prove the bound χ(G2) ≤ ⌈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(G2) ≤ ⌈5/3 Δ⌉ + 25. We generalize this result to L(p, q)-labeling of planar graphs, by showing that λqp(G) ≤ q ⌈5/3 Δ⌉ + 18p + 77q - 18. For each of the results, the proof provides a quadratic time algorithm.

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