Perfect colourings of regular graphs

TL;DR

This paper characterizes perfect colourings of regular graphs, providing necessary and sufficient conditions for their colour incidence matrices, and classifies all such colourings for certain regular graphs and Platonic solids.

Contribution

It establishes new necessary and sufficient conditions for perfect colourings and classifies all perfect colourings for specific regular graphs and Platonic solids.

Findings

Characterization of perfect colourings via incidence matrices

Complete classification for 3-, 4-, and 5-regular graphs with limited colours

Identification of perfect colourings of Platonic solid edge graphs

Abstract

A vertex colouring of some graph is called perfect if each vertex of colour $i$ has exactly $a_{ij}$ neighbours of colour $j$. Being perfect imposes several restrictions on the colour incidence matrix $(a_{ij})$. We list several (old and new) necessary conditions for a matrix to be the colour incidence matrix of a perfect colouring. Moreover we show that a certain combination of these conditions is also sufficient. Using this we determine a list of all colour incidence matrices corresponding to perfect colourings of 3-regular, 4-regular and 5-regular graphs with two, three and four colours, respectively. As an application we determine all perfect colourings of the edge graphs of the Platonic solids with two, three and four colours, respectively.