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arXiv · 2206.00900

The chromatic index of finite projective spaces

Lei Xu·Tao Feng
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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 $q\in\{3,4,8,16\}$.

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Lei Xu, Tao Feng. 2022-06-02. The chromatic index of finite projective spaces. https://arxiv.org/abs/2206.00900

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