Johan Colouring of Graph Operations

TL;DR

This paper characterizes graphs that admit a Johan colouring, a special proper coloring where each vertex's closed neighborhood contains all color classes, and explores conditions under which graph operations preserve this property.

Contribution

It provides a characterization of graphs with Johan colourings and investigates how certain graph operations affect the existence of Johan colourings.

Findings

Graphs with Johan colourings are characterized.

Conditions under which graph operations preserve Johan colourings are identified.

Preliminary results on graph operations and Johan colourings are discussed.

Abstract

A vertex $v$ of a given graph is said to be in a rainbow neighbourhood of $G$ if every colour class of $G$ consists of at least one vertex from the closed neighbourhood $N[v]$. A maximal proper colouring of a graph $G$ is a Johan colouring if and only if every vertex of $G$ belongs to a rainbow neighbourhood of $G$. In general all graphs need not have a Johan colouring, even though they admit a chromatic colouring. In this paper, we characterise graphs which admit a Johan colouring. We also discuss some preliminary results in respect of certain graph operations which admit a Johan colouring under certain conditions.