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

Maurice Genevieva Almeida

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

Combinatorics 2025-09-10 v3

Authors: Maurice Genevieva Almeida

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.

Keywords

graph coloring rainbow coloring ramsey theory and colorings

Cite

@article{arxiv.2505.07758,
  title  = {$(2k+1)$-Neighborhood Balanced Coloring},
  author = {Maurice Genevieva Almeida},
  journal= {arXiv preprint arXiv:2505.07758},
  year   = {2025}
}

Comments

The general version of neighborhood balanced k-coloring is uploaded as a separate content with work from four different authors (arXiv:2509.06003)

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