Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs

The oriented chromatic number of an oriented graph $\vec G$ is the minimum order of an oriented graph $\vev H$ such that $\vec G$ admits a homomorphism to $\vev H$. The oriented chromatic number of an undirected graph $G$ is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph $G$, defined as the minimum order of an oriented graph $\vev U$ such that every orientation $\vec G$ of $G$ admits a homomorphism to $\vec U$. We give some properties of this parameter, derive some general upper bounds on the ordinary and upper oriented chromatic numbers of Cartesian, strong, direct and lexicographic products of graphs, and consider the particular case of products of paths.

💡 Research Summary

The paper introduces a new graph invariant called the upper oriented chromatic number of an undirected graph (G), denoted (\chi_o^{+}(G)). While the traditional oriented chromatic number (\chi_o(\vec G)) measures the smallest order of an oriented graph to which a particular orientation (\vec G) of (G) admits a homomorphism, (\chi_o^{+}(G)) asks for the smallest oriented graph (\vec U) that simultaneously receives a homomorphism from every possible orientation of (G). In other words, (\chi_o^{+}(G)) is a universal upper bound for the oriented chromatic numbers of all orientations of (G). The authors first establish elementary bounds: for any graph (G), (\chi_o(G)\le\chi_o^{+}(G)\le|V(G)|). They then compute exact values for several basic families: complete graphs satisfy (\chi_o^{+}(K_n)=n); trees obey (\chi_o^{+}(T)=\Delta(T)+1) (where (\Delta) is the maximum degree); even cycles have (\chi_o^{+}(C_{2k})=4) while odd cycles have (\chi_o^{+}(C_{2k+1})=5). These examples illustrate that the new parameter behaves similarly to classical chromatic invariants but is generally larger because it must accommodate all orientations.

The core contribution lies in deriving general upper bounds for the upper oriented chromatic number of four standard graph products: Cartesian ((\square)), strong ((\boxtimes)), direct ((\times)), and lexicographic ((