On J-Colorability of Certain Derived Graph Classes

TL;DR

This paper investigates the properties of J-colorings, a type of maximal proper coloring where each vertex's neighborhood contains all colors, focusing on specific Mycielski-type graphs.

Contribution

It introduces and analyzes parameters related to J-coloring in certain derived graphs, expanding understanding of rainbow neighborhoods in graph theory.

Findings

Characterization of J-colorability in Mycielski-type graphs

Identification of parameters influencing rainbow neighborhoods

Extension of J-coloring concepts to new graph classes

Abstract

A vertex $v$ of a given graph $G$ is said to be in a rainbow neighbourhood of $G$, with respect to a proper coloring $C$ of $G$, if the closed neighbourhood $N[v]$ of the vertex $v$ consists of at least one vertex from every colour class of $G$ with respect to $C$. A maximal proper colouring of a graph $G$ is a $J$-colouring of $G$ if and only if every vertex of G belongs to a rainbow neighbourhood of $G$. In this paper, we study certain parameters related to $J$-colouring of certain Mycielski type graphs.