Graph unique-maximum and conflict-free colorings

TL;DR

This paper explores the relationship between unique-maximum and conflict-free vertex colorings in graphs, analyzing their properties, computational complexity, and improving bounds for grid graphs.

Contribution

It establishes connections between two coloring types, studies their computational complexity, and enhances lower bounds for grid graph chromatic numbers.

Findings

Proves a completeness result for conflict-free coloring complexity

Establishes relationships between unique-maximum and conflict-free colorings

Improves lower bounds for grid graph chromatic numbers

Abstract

We investigate the relationship between two kinds of vertex colorings of graphs: unique-maximum colorings and conflict-free colorings. In a unique-maximum coloring, the colors are ordered, and in every path of the graph the maximum color appears only once. In a conflict-free coloring, in every path of the graph there is a color that appears only once. We also study computational complexity aspects of conflict-free colorings and prove a completeness result. Finally, we improve lower bounds for those chromatic numbers of the grid graph.