Proper conflict-free and unique-maximum colorings of planar graphs with respect to neighborhoods

TL;DR

This paper studies conflict-free and unique-maximum colorings of planar graphs with respect to neighborhoods, establishing bounds on the number of colors needed for proper and improper colorings, with a focus on planar and outerplanar graphs.

Contribution

It introduces the study of proper conflict-free and unique-maximum colorings in planar graphs, providing upper bounds and constructions for the chromatic numbers in these settings.

Findings

Every planar graph admits a proper unique-maximum coloring with at most 10 colors.

There exist planar graphs requiring at least 6 colors for such colorings.

Tight upper bounds are established for outerplanar graphs.

Abstract

A {\em conflict-free coloring} of a graph {\em with respect to open} (resp., {\em closed}) {\em neighborhood} is a coloring of vertices such that for every vertex there is a color appearing exactly once in its open (resp., closed) neighborhood. Similarly, a {\em unique-maximum coloring} of a graph {\em with respect to open} (resp., {\em closed}) {\em neighborhood} is a coloring of vertices such that for every vertex the maximum color appearing in its open (resp., closed) neighborhood appears exactly once. There is a vast amount of literature on both notions where the colorings need not be proper, i.e., adjacent vertices are allowed to have the same color. In this paper, we initiate a study of both colorings in the proper settings with the focus given mainly to planar graphs. We establish upper bounds for the number of colors in the class of planar graphs for all considered colorings and provide constructions of planar graphs attaining relatively high values of the corresponding chromatic numbers. As a main result, we prove that every planar graph admits a proper unique-maximum coloring with respect to open neighborhood with at most 10 colors, and give examples of planar graphs needing at least $6$ colors for such a coloring. We also establish tight upper bounds for outerplanar graphs. Finally, we provide several new bounds also for the improper setting of considered colorings.