# A note on chromatic blending of colour clusters

## TL;DR

This paper explores a graph-theoretic model of colour clusters, focusing on maximizing edges while maintaining a fixed chromatic number, and introduces a recursive blending process to achieve total chromatic blending, applicable to various fields.

## Contribution

It introduces a novel graph structure for colour clusters and a recursive blending process to model total chromatic blending across different disciplines.

## Key findings

- Defined a graph model with maximum edges for given colour clusters.
- Developed a recursive blending process to achieve total chromatic blending.
- Illustrated applications in genetics, chemistry, and social sciences.

## Abstract

For a colour cluster $\C =(\mathcal{C}_1,\mathcal{C}_2, \mathcal{C}_3,\dots,\mathcal{C}_\ell)$, $\mathcal{C}_i$ is a colour class, and $|\mathcal{C}_i|=r_i \geq 1$, we investigate a simple connected graph structure $G^{\C}$, which represents a graphical embodiment of the colour cluster such that the chromatic number $\chi(G^{\C})= \ell,$ and the number of edges is a maximum, denoted $\varepsilon^+(G^{\C})$. We also extend the study by inducing new colour clusters recursively by blending the colours of all pairs of adjacent vertices. Recursion repeats until a maximal homogeneous blend between all $\ell$ colours is obtained. This is called total chromatic blending. Total chromatic blending models for example, total genetic, chemical, cultural or social orderliness integration.