Neighborhood Balanced k-Coloring of Graphs

Maurice Genevieva Almeida; Tarkeshwar Singh; Siddharth Gupta; Ravindra Pawar

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arXiv · 2509.06003

Neighborhood Balanced k-Coloring of Graphs

Maurice Genevieva Almeida·Tarkeshwar Singh·Siddharth Gupta·Ravindra Pawar

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Abstract

For a simple graph G = (V, E) and a positive integer k greater than or equal to 2, a coloring of vertices of G using exactly k colors such that each vertex has an equal number of neighbors of each color is called neighborhood-balanced k-coloring, and the graph is called a neighborhood-balanced k-colored graph. This generalizes the notion of neighborhood balanced coloring of graphs introduced by Bryan Freyberg and Alison Marr (Graphs and Combinatorics, 2024). We derive some necessary/sufficient conditions for a graph to admit a neighborhood-balanced k-coloring and discuss several graph classes that admit such colorings. We also show that the problem of determining whether a given graph has such a coloring is NP-complete. Furthermore, we prove that there is no forbidden subgraph characterization for the class of neighborhood-balanced k-colorable graphs.

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Maurice Genevieva Almeida, Tarkeshwar Singh, Siddharth Gupta, Ravindra Pawar. 2026-08-19. Neighborhood Balanced k-Coloring of Graphs. https://arxiv.org/abs/2509.06003

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