$(2k+1)$-Neighborhood Balanced Coloring

TL;DR

This paper introduces the concept of $(2k+1)$-neighborhood balanced coloring in graphs, analyzing its properties and identifying graph families that admit such colorings, with implications for graph symmetry and coloring problems.

Contribution

It defines and explores the properties of $(2k+1)$-neighborhood balanced colorings, providing new results for specific graph families and expanding understanding of equitable colorings.

Findings

Certain graph families admit $(2k+1)$-neighborhood balanced colorings

Characterization results for graphs with this coloring property

Examples of graphs with and without the property

Abstract

Let $G=(V,E)$ be a simple graph and $(2k+1)$ be a prime integer. Let each vertex of $G$ be colored using one of the $(2k+1)$ colors, say $R_1,R_2,...,R_{2k+1}$. If every vertex has an equal number of neighbors of each color, then the coloring is a $(2k+1)$-neighborhood balanced coloring. We establish a number of results for common families of graphs and present some families of graphs that have this property.