# Determining the b-chromatic number of subdivision-vertex neighbourhood coronas

## TL;DR

This paper determines the b-chromatic number for specific graph constructions called subdivision-vertex neighbourhood coronas involving paths, cycles, stars, and complete graphs, expanding understanding of graph coloring properties.

## Contribution

It provides exact b-chromatic numbers for subdivision-vertex neighbourhood coronas of paths, cycles, stars, and certain complete graphs, with illustrative proofs and examples.

## Key findings

- Exact b-chromatic numbers for G⊙H where G,H are paths, cycles, or stars.
- Results for K_n⊙G with degree constraints.
- Analysis of K_n⊙G where degree conditions are met.

## Abstract

Let $G$ and $H$ be two graphs, each one of them being a path, a cycle or a star. In this paper, we determine the $b$-chromatic number of every subdivision-vertex neighbourhood corona $G\boxdot H$ or $G\boxdot K_n$, where $K_n$ is the complete graph of order $n$. It is also established for those graphs $K_n\boxdot G$ having $m$-degree not greater than $n+2$. All the proofs are accompanied by illustrative examples.