Achromatic arboricity on complete graphs

Abstract

AbstractIn this paper we study the 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 $$A_{\alpha }(G)$$ A α ( 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 $$\begin{aligned} \frac{1}{4}n^{\frac{3}{2}}-c_1n \le A_{\alpha }(G)\le \frac{1}{\sqrt{2}}n^{\frac{3}{2}}-c_2n \end{aligned}$$ 1 4 n 3 2 - c 1 n ≤ A α ( G ) ≤ 1 2 n 3 2 - c 2 n where $$c_1$$ c 1 and $$c_2$$ c 2 are constants such that $$0<c_1,c_2<\frac{1}{2}$$ 0 < c 1 , c 2 < 1 2 .