On chromatic indices of finite affine spaces

Gabriela Araujo-Pardo, György Kiss, Christian Rubio-Montiel, Adrián Vázquez-Ávila

Published 2017-11-24Version 1

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.