A $p$-centered coloring for the grid using $O(p)$ colors

TL;DR

This paper presents an explicit method to color the planar grid with O(p) colors, ensuring that every connected subgraph either has a unique-colored vertex or more than p colors, aiding graph analysis.

Contribution

It provides the first explicit construction of p-centered colorings for the planar grid with a linear number of colors in p.

Findings

Constructed p-centered coloring with O(p) colors for the planar grid

Ensured every connected subgraph has a unique color or more than p colors

Facilitated analysis of planar grid subgraphs

Abstract

A $p$-centered coloring of a graph $G$, where $p$ is a positive integer, is a coloring of the vertices of $G$ in such a way that every connected subgraph of $G$ either contains a vertex with a unique color or contains more than $p$ different colors. We give an explicit construction of a $p$-centered coloring using $O(p)$ colors for the planar grid.