Achromatic arboricity on complete graphs

TL;DR

This paper investigates the maximum number of edge colors in complete graphs where each color class forms a forest and any two classes contain cycles, establishing bounds proportional to n^{3/2}.

Contribution

It introduces the concept of achromatic arboricity for complete graphs and derives asymptotic bounds for this parameter.

Findings

Achromatic arboricity of complete graphs is bounded between rac{1}{4}n^{3/2} and rac{1}{ oot 2 ext{}} n^{3/2}.

The bounds are tight up to a constant factor, providing new insights into edge colorings with cycle constraints.

The work extends arboricity concepts to a new achromatic setting with cycle interactions.

Abstract

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 \[\frac{1}{4}n^{\frac{3}{2}}-\Theta(n) \leq A_{\alpha}(G)\leq \frac{1}{\sqrt{2}}n^{\frac{3}{2}}-\Theta(n).\]