Closed Neighborhood Balanced k-Coloring of Graphs

TL;DR

This work introduces closed neighborhood balanced $k$-coloring, a generalization of two-color closed neighborhood balance to $k$ colors, and studies its structural and computational properties. It derives counting constraints that tie color-class sizes to edge distributions, demonstrates the non-hereditary nature of CNBC graphs, and analyzes how CNBC behaves under standard graph operations, including complementation, products, and joins. The authors also show that CNBC is NP-hard for all $k\ge3$ via a reduction from $k$-coloring, and they present constructions that yield CNBC graphs with unbalanced color classes, underscoring the richness and complexity of the class. Open questions remain regarding a complete characterization of regular CNBC graphs and the precise complexity boundary at $k=2$, inviting further investigation into the combinatorial structure of closed neighborhood colorings.

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 every vertex has an equal number of vertices of each color in its closed neighborhood is called closed neighborhood-balanced k-coloring, and the graph which admits such a coloring is called closed neighborhood balanced k-colored graph. We derive some necessary/sufficient conditions for a graph to admit a closed neighborhood balanced k-coloring and discuss various graph operations involving such graphs. Furthermore, we prove that there is no forbidden subgraph characterization for the class of closed neighborhood-balanced k-colorable graphs.

Figures (1)

• Figure 1: A closed neighborhood-balanced $2$-coloring of $K_{6}$.

Theorems & Definitions (46)

• Definition 1.1
• Lemma 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6