Connectedness of Strong k-Colour Graphs

TL;DR

This paper investigates the connectivity of the strong k-colour graph S_k(G), which represents proper vertex colourings of a graph G with k colours, focusing on when S_k(G) is connected.

Contribution

The paper provides new results characterizing the conditions under which the strong k-colour graph S_k(G) is connected for various graphs G and values of k.

Findings

Identifies conditions for connectivity of S_k(G)

Characterizes graphs G with connected S_k(G) for specific k

Provides theoretical insights into colourings and graph structure

Abstract

For a positive integer k and a graph G, we consider proper vertex-colourings of G with k colours in which all k colours are actually used. We call such a colouring a strong k-colouring. The strong k-colour graph of G, S_k(G), is the graph that has all the strong k-colourings of G as its vertex set, and two colourings are adjacent in S_k(G) if they differ in colour on only one vertex of G. In this paper, we show some results related to the question : For what G and k is S_k(G) connected ?