Acyclic chromatic index of planar graphs with triangles

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G  , denoted by χa′(G), is the least number of colors in an acyclic edge coloring of G. Let G   be a planar graph with maximum degree Δ(G)Δ(G). In this paper, we show that χa′(G)⩽Δ(G)+4, if G   contains no 4-cycle; χa′(G)⩽Δ(G)+5, if G   contains no intersecting triangles; and χa′(G)⩽Δ(G)+6 if G contains no adjacent triangles.

► We study acyclic edge coloring of planar graphs with maximum degree D  .
► Planar graphs without 4-cycles are acyclically (D+4)(D+4)-colorable.
► Planar graphs without intersecting triangles are acyclically (D+5)(D+5)-colorable.
► Planar graphs without adjacent triangles are acyclically (D+6)(D+6)-colorable.