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

Abstract

Let the vertices of a Cartesian product graph G H be ordered by an ordering σ. By the First-Fit coloring of (G H, σ) we mean the vertex coloring procedure which scans the vertices according to the ordering σ and for each vertex assigns the smallest available color. Let F F (G H, σ) be the number of colors used in this coloring. By introducing the concept of descent we obtain a sufficient condition to determine whether F F (G H, σ) = F F (G H, τ ), where σ and τ are arbitrary orders. We study and obtain some bounds for F F (G H, σ), where σ is any quasi-lexicographic ordering. The First-Fit coloring of (G H, σ) does not always yield an optimum coloring. A greedy defining set of (G H, σ) 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 H, σ) yields an optimum coloring. We show that the First-Fit coloring and greedy defining sets of G 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 H, including some extremal results for Latin squares.