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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.

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) ≤ ∆ + 2 for any simple graph G with maximum degree ∆. In this paper, we prove that if G is a planar graph, then a ′ (G) ≤ ∆ + 7. This improves a result by Basavaraju et al. [Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math., 25 (2011), pp. 463-478], which says that every planar graph G satisfies a ′ (G) ≤ ∆ + 12.

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a b s t r a c tAn 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 k such that G has an acyclic edge coloring using k colors.In this paper, we prove that every planar graph G with girth g(G) and maximum degree ∆ has a ′ (G) = ∆ if there exists a pair (k, m) ∈ {(3, 11), (4, 8), (5, 7), (8, 6)} such that G satisfies ∆ ≥ k and g(G) ≥ m.