A new upper bound on the acyclic chromatic indices of planar graphs

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a ′ ( G ) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. It was conjectured that a ′ ( G ) ≤ Δ + 2 for any simple graph G with maximum degree Δ . In this paper, we prove that if G is a planar graph, then a ′ ( G ) ≤ Δ + 7 . This improves a result by Basavaraju et al. [M. Basavaraju, L.S. Chandran, N. Cohen, F. Havet, T. Müller, Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2011) 463–478], which says that every planar graph G satisfies a ′ ( G ) ≤ Δ + 12 .