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

TL;DR

This paper investigates the First-Fit coloring process for Cartesian product graphs, introduces the concept of descent to analyze coloring bounds, and explores greedy defining sets to achieve optimal colorings, with applications to Latin squares.

Contribution

It introduces the concept of descent for analyzing First-Fit coloring and establishes bounds and properties of greedy defining sets in Cartesian product graphs.

Findings

Descent is a sufficient condition for coloring equivalence.

Bounds for First-Fit coloring numbers are established.

Greedy defining sets coincide with First-Fit colorings for quasi-lexicographic orders.

Abstract

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.