The chromatic index of finite projective spaces

A line coloring of PG(n,q) $\text{PG}(n,q)$, the n $n$-dimensional projective space over GF(q) $(q)$, is an assignment of colors to all lines of PG(n,q) $\text{PG}(n,q)$ so that any two lines with the same color do not intersect. The chromatic index of PG(n,q) $\text{PG}(n,q)$, denoted by & chi;& PRIME;(PG(n,q)) $\chi <^>{\prime} (\text{PG}(n,q))$, is the least number of colors for which a coloring of PG(n,q) $\text{PG}(n,q)$ exists. This paper translates the problem of determining the chromatic index of PG(n,q) $\text{PG}(n,q)$ to the problem of examining the existences of PG(3,q) $\text{PG}(3,q)$ and PG(4,q) $\text{PG}(4,q)$ with certain properties. In particular, it is shown that for any odd integer n $n$ and q & ISIN;{3,4,8,16},& chi;& PRIME;(PG(n,q))=(qn-1)/(q-1) $q\in \{3,4,8,16\},\chi <^>{\prime} (\text{PG}(n,q))=({q}<^>{n}-1)\unicode{x02215}(q-1)$, which implies the existence of a parallelism of PG(n,q) $\text{PG}(n,q)$ for any odd integer n $n$ and q & ISIN;{3,4,8,16} $q\in \{3,4,8,16\}$.