The chromatic index of finite projective spaces

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