Acyclic edge coloring of planar graphs without a $$3$$3-cycle adjacent to a $$6$$6-cycle

An acyclic edge coloring of a graph $$G$$G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $$a'(G)$$a′(G) of $$G$$G is the smallest integer $$k$$k such that $$G$$G has an acyclic edge coloring using $$k$$k colors. Fiamč ik (Math Slovaca 28:139–145, 1978) and later Alon et al. (J Graph Theory 37:157–167, 2001) conjectured that $$a'(G)\le \Delta +2$$a′(G)≤Δ+2 for any simple graph $$G$$G with maximum degree $$\Delta $$Δ. In this paper, we confirm this conjecture for planar graphs without a $$3$$3-cycle adjacent to a $$6$$6-cycle.