The time complexity of oriented chromatic number for acyclic oriented connected subcubic subgraphs of grids

Abstract

An oriented $k$-coloring of an oriented graph $\vec{G}=(V,A)$ is a partition of $V$ into $k$ color classes, such that there is no pair of adjacent vertices belonging to the same class and all the arcs between a pair of classes have the same orientation. The smallest $k$ such that $\vec{G}$ admits an oriented $k$-coloring is the oriented chromatic number ${\chi }_o\left(\vec{G}\right)$ of $\vec{G}$. The $k$-oriented chromatic number decision problem asks whether an oriented graph $\vec{G}$ has oriented chromatic number at most $k$. $k$-oriented chromatic number is a polynomial problem, when $k\le 3$ and $NP$-complete, when $k\ge 4$ even if input $\vec{G}$ is acyclic and the underlying graph $G$ is bipartite, cubic and planar. In 2003, Fertin, Raspaud and Roychowdhury established exact values and bounds on the oriented chromatic number for several grid subgraph classes. But, the time complexity of $k$-oriented chromatic number was unknown for grid subgraphs. In this work we prove that the $k$-oriented chromatic number problem is NP-complete even if $\vec{G}$ is acyclic oriented, $k\ge 4$, and the underlying graph $G$ is a connected and subcubic subgraph of a grid.