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

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 be ordered by an ordering . By the First-Fit coloring of we mean the vertex coloring procedure which scans the vertices according to the ordering and for each vertex assigns the smallest available color. Let be the number of colors used in this coloring. By introducing the concept of descent we obtain a sufficient condition to determine whether , where and are arbitrary orders. We study and obtain some bounds for , where is any quasi-lexicographic ordering. The First-Fit coloring of does not always yield an optimum coloring. A greedy defining set of is a subset of vertices in the graph together with a suitable pre-coloring of such that by fixing…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
