The chromatic index of finite projective spaces

Abstract

A line coloring of $\text{PG}\left(n,q\right)$, the $n$-dimensional projective space over GF$\left(q\right)$, is an assignment of colors to all lines of $\text{PG}\left(n,q\right)$ so that any two lines with the same color do not intersect. The chromatic index of $\text{PG}\left(n,q\right)$, denoted by $\chi \prime \left(\text{PG}\left(n,q\right)\right)$, is the least number of colors for which a coloring of $\text{PG}\left(n,q\right)$ exists. This paper translates the problem of determining the chromatic index of $\text{PG}\left(n,q\right)$ to the problem of examining the existences of $\text{PG}\left(3,q\right)$ and $\text{PG}\left(4,q\right)$ with certain properties. In particular, it is shown that for any odd integer $n$ and $q\in \left\{3,4,8,16\right\},\chi \prime \left(\text{PG}\left(n,q\right)\right)=\left(q^n-1\right)∕\left(q-1\right)$, which implies the existence of a parallelism of $\text{PG}\left(n,q\right)$ for any odd integer $n$ and $q\in \left\{3,4,8,16\right\}$.