On Certain $J$-Colouring Parameters of Graphs

TL;DR

This paper introduces Johan colouring, a new graph colouring concept inspired by rainbow neighbourhoods, providing bounds and explicit results for various graph classes.

Contribution

It proposes Johan colouring, explores its properties, and establishes bounds and explicit results for specific graph families, advancing graph colouring theory.

Findings

Upper bound for connected graphs established

Explicit Johan colouring results for cycles and complete graphs

Insights into maximal versus minimal colouring approaches

Abstract

In this paper, a new type of colouring called Johan colouring is introduced. This colouring concept is motivated by the newly introduced invariant called the rainbow neighbourhood number of a graph. The study ponders on maximal colouring opposed to minimum colouring. An upper bound for a connected graph is presented and a number of explicit results are presented for cycles, complete graphs, wheel graphs and for a complete $l$-partite graph.