Achromatic arboricity on complete graphs

In this paper we study the {\it {achromatic arboricity}} of the complete graph. This parameter arises from the arboricity of a graph as the achromatic index arises from the chromatic index. The achromatic arboricity of a graph $G$, denoted by $A_{\alpha}(G)$, is the maximum number of colors that can be used to color the edges of $G$ such that every color class induces a forest but any two color classes contain a cycle. In particular, if $G$ is a complete graph we prove that $$\frac{1}{4}n^{\frac{3}{2}}-\Theta(n) \leq A_{\alpha}(G)\leq \frac{1}{\sqrt{2}}n^{\frac{3}{2}}-\Theta(n).$$