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

TL;DR

The paper investigates arc colorings in three finite geometries: the projective plane $\text{PG}(2,q)$, the affine plane $\text{AG}(2,q)$, and Euclidean grids $G_n$, focusing on the arc chromatic number $\chi_{\mathcal{A}}$ and its fractional variant $\chi_{\mathcal{A},f}$. It proves the exact value $\chi_{\mathcal{A}}(\text{PG}(2,q)) = q+1$ for all prime powers $q$ using a cyclic-model coloring and a double-counting argument, and determines $\chi_{\mathcal{A},f}(\text{PG}(2,q))$ with parity-based formulas: $\frac{q^2+q+1}{q+1}$ for odd $q$ and $\frac{q^2+q+1}{q+2}$ for even $q$, illustrating a sharp distinction between regular and fractional chromatic numbers. For $\text{AG}(2,q)$, the exact chromatic number is $q$ for odd $q$, and $q-1$ or $q$ for $q=2^k$ with $k\ge4$, with explicit arc-proper $(q-1)$-colorings found for $q=2,4,8$ (the latter two via a Daisy-Structure analysis and computation). In the Euclidean grid, general bounds $\frac{n}{2} \le \chi_{\mathcal{A}}(G_n) \le 2n$ are given, together with an asymptotic bound $\chi_{\mathcal{A}}(G_n) \le (1+\varepsilon)n$ for large $n$ and exact colorings for small $n$ (notably $n=4,5,8$); the $n=9$ case remains open between $5$ and $6$. The paper closes with a set of open problems and directions, including the fractional and exact arc chromatic numbers for other regimes and geometries, highlighting rich directions for future research.

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 $(1+ε)n$ sets in general position exists for any $ε> 0$ and sufficiently large $n$. We also present exact minimal partitions for small Euclidean grids.

Figures (3)

• Figure 1: The Daisy Structure of $\text{PG}(2,4)$ (left). The structure of $\text{AG}(2,4)$ (right).
• Figure 2: An arc-chromatic $3$-coloring for $\text{AG}(2,4)$.
• Figure 3: An arc-proper $4$-coloring of $G_8$.

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1
• proof
• Proposition 2.2
• proof