The Oriented Chromatic Number of the Hexagonal Grid is 6

TL;DR

This paper determines that the oriented chromatic number of the hexagonal grid family is exactly 6, resolving a previous gap between known bounds.

Contribution

The paper proves that the lower bound for the oriented chromatic number of hexagonal grids is 6, establishing the exact value.

Findings

The oriented chromatic number of hexagonal grids is exactly 6.

Previous bounds were improved to a precise value.

The result closes the gap in the known bounds.

Abstract

The oriented chromatic number of a directed graph $G$ is the minimum order of an oriented graph to which $G$ has a homomorphism. The oriented chromatic number $\chi_o({\cal F})$ of a graph family ${\cal F}$ is the maximum oriented chromatic number over any orientation of any graph in ${\cal F}$. For the family of hexagonal grids ${\cal H}_2$, Bielak (2006) proved that $5 \le \chi_o({\cal H}_2) \le 6$. Here we close the gap by showing that $\chi_o({\cal H}_2) \ge 6$.