The chromatic index of finite projective spaces

TL;DR

This paper determines the chromatic index of finite projective spaces PG(n,q) for certain parameters, linking it to the existence of specific geometric configurations and establishing new parallelisms.

Contribution

It translates the problem of finding the chromatic index into geometric existence problems and proves the index for odd n and specific q values, confirming new parallelisms.

Findings

For odd n and q in {3,4,8,16}, the chromatic index is (q^n-1)/(q-1).

Establishes the existence of parallelisms in PG(n,q) for these parameters.

Links coloring problems to geometric configurations in finite projective spaces.

Abstract

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 $q\in\{3,4,8,16\}$.