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

Abstract

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. Fiamă? 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.