First-Fit coloring of Cartesian product graphs and its defining sets

Let the vertices of a Cartesian product graph $G\Box H$ be ordered by an ordering $\sigma$. By the First-Fit coloring of $(G\Box H, \sigma)$ we mean the vertex coloring procedure which scans the vertices according to the ordering $\sigma$ and for each vertex assigns the smallest available color. Let $FF(G\Box H,\sigma)$ be the number of colors used in this coloring. By introducing the concept of descent we obtain a sufficient condition to determine whether $FF(G\Box H,\sigma)=FF(G\Box H,\tau)$, where $\sigma$ and $\tau$ are arbitrary orders. We study and obtain some bounds for $FF(G\Box H,\sigma)$, where $\sigma$ is any quasi-lexicographic ordering. The First-Fit coloring of $(G\Box H, \sigma)$ does not always yield an optimum coloring. A greedy defining set of $(G\Box H, \sigma)$ is a subset $S$ of vertices in the graph together with a suitable pre-coloring of $S$ such that by fixing the colors of $S$ the First-Fit coloring of $(G\Box H, \sigma)$ yields an optimum coloring. We show that the First-Fit coloring and greedy defining sets of $G\Box H$ with respect to any quasi-lexicographic ordering (including the known lexicographic order) are all the same. We obtain upper and lower bounds for the smallest cardinality of a greedy defining set in $G\Box H$, including some extremal results for Latin squares.