Let G=(V,E) be any finite graph. A mapping c:E→[k] is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G, denoted by ~(χ′)(G). In 1978, Fiamk conjectured that for any graph G it holds ~(χ′)(G)≤Δ(G)+2, where Δ(G) stands for the maximum degree of G. This conjecture has been verified by now only for some special classes of graphs. In 2010, Borowiecki and Fiedorowicz confirmed it for planar graph with girth at least 5. In this paper, we improve the above result, by proving that if G is a plane graph such that for each pair i,j∈3,4, no i-face and a j-face share a common vertex in G, then ~(χ′)(G)≤Δ(G)+2.
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