On chromatic indices of finite affine spaces

TL;DR

This paper investigates the maximum number of colors in line-colorings of finite affine spaces, establishing asymptotic formulas for pseudoachromatic and achromatic indices depending on the dimension's parity.

Contribution

It provides asymptotic bounds for the pseudoachromatic index and exact values for the achromatic index of finite affine spaces, especially in the planar case.

Findings

Pseudoachromatic index grows as q^{1.5n-1} for even n.

Bounds for odd n are established between q^{1.5(n-1)} and q^{1.5n-1}.

Exact values are obtained for the planar case.

Abstract

The pseudoachromatic index of the finite affine space $\mathrm{AG}(n,q),$ denoted by $\psi'(\mathrm{AG}(n,q)),$ is the the maximum number of colors in any complete line-coloring of $\mathrm{AG}(n,q).$ When the coloring is also proper, the maximum number of colors is called the achromatic index of $\mathrm{AG}(n,q).$ We prove that if $n$ is even then $\psi'(\mathrm{AG}(n,q))\sim q^{1.5n-1}$; while when $n$ is odd the value is bounded by $q^{1.5(n-1)}<\psi'(\mathrm{AG}(n,q))<q^{1.5n-1}$. Moreover, we prove that the achromatic index of $\mathrm{AG}(n,q)$ is $q^{1.5n-1}$ for even $n,$ and we provides the exact values of both indices in the planar case.