Bounds and monotonicity of critical set parameters of colourings

Abstract

Given a graph G and a k-colouring c, a critical set S is a minimal set of vertices S for which the restriction of c to S determines c completely. There are four parameters that relate to the size of a critical set in a graph colouring problem. We prove that each of these are upper bounded by n−1 and equality can be attained. For three of these parameters, we present some simple characterizations of the graphs attaining the maximum value n−1. As a second main result, we answer a question by Cooper and Kirkpatrick, showing that there is monotone behaviour in the number of colours for only two of the four parameters. We investigate the monotone behaviour for the subgraph-order as well.