Improved bounds for acyclic chromatic index of planar graphs

Acyclic coloring problem is a specialized problem that arises in the efficient computation of Hessians. A proper edge coloring of a graph GG is called acyclic if there is no 2-colored cycle in GG. The acyclic chromatic index of GG, denoted by χa′(G), is the least number of colors in an acyclic edge coloring of GG. Let GG be planar graphs with girth gg and maximum degree ΔΔ. In this paper, it is shown that if g≥4g≥4 and Δ≥8Δ≥8, then χa′(G)≤Δ+3; if g≥5g≥5 and Δ≥10Δ≥10 or g≥6g≥6 and Δ≥6Δ≥6, then χa′(G)=Δ.