Acyclic chromatic indices of fully subdivided graphs

Let G=(V,E) be any finite simple graph. A mapping @f: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 and is denoted by @g"a^'(G). Fiamcik proved that @D(G)@?(@D(G)-1)+1 is an upper bound for the acyclic chromatic index of a graph G and conjectured that @g"a^'(G)= =3 or G is acyclic. Our proof is constructive and hence yields a polynomial time algorithm. This result is an extension of a result obtained in 1984 by Fiamcik for subdivisions of cubic graphs and improves a recent result presented by Muthu et al.