The arc chromatic number for Galois projective planes, affine planes and Euclidean grids

TL;DR

This paper determines the minimum number of arcs needed to partition various finite geometries like Galois projective and affine planes, and extends results to Euclidean grids, providing exact values and bounds.

Contribution

It establishes exact arc partition numbers for Galois projective planes and affine planes, and extends these findings to Euclidean grids, including bounds for specific cases.

Findings

Minimum arcs for Galois projective plane is q+1

Affine planes can be partitioned into q arcs, tight for odd prime powers

Partition into (1+ε)n sets in Euclidean grids for large n

Abstract

We establish that the minimum number of arcs required to partition the Galois projective plane $\text{PG}(2,q)$ is $q+1$. Furthermore, we determine the exact value for a fractional variant of this problem. We extend our analysis to affine planes $\text{AG}(2,q)$, proving that they can be partitioned into $q$ arcs. In particular, we show that this partition is tight when $q$ is an odd prime power, and that a $(q-1)$-partition is attainable for $q=2^k$ with $k \in \{1,2,3\}$. For $q=2^k$ with $k \geq 4$, we provide bounds between two possible values. Finally, we apply these results to Euclidean grids, demonstrating that a partition into $(1+\epsilon)n$ sets in general position exists for any $\epsilon > 0$ and sufficiently large $n$. We also present exact minimal partitions for small Euclidean grids.