2008) 'Connectedness of the graph of vertex-colourings.', Discrete mathematics., 308 (5-6). pp. 913-919. Further information on publisher's website: http://dx. Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractFor a positive integer k and a graph G, the k-colour graph of G, C k (G), is the graph that has the proper k-vertex-colourings of G as its vertex set, and two k-colourings are joined by an edge in C k (G) if they differ in colour on just one vertex of G. In this note some results on the connectivity of C k (G) are proved. In particular it is shown that if G has chromatic number k ∈ {2, 3}, then C k (G) is not connected. On the other hand, for k ≥ 4 there are graphs with chromatic number k for which C k (G) is not connected, and there are k-chromatic graphs for which C k (G) is connected.

Given a 3-colorable graph G together with two proper vertex 3-colorings and of G, consider the following question: is it possible to transform into by recoloring vertices of G one at a time, making sure that all intermediate colorings are proper 3-colorings? We prove that this question is answerable in polynomial time. We do so by characterizing the instances G, , where the transformation is possible; the proof of this characterization is via an algorithm that either finds a sequence of recolorings between and , or exhibits a structure which proves that no such

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.

The generalised colouring numbers col r (G) and wcol r (G) were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications.In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the r-colouring number col r and a polynomial bound for the weak r-colouring number wcol r . In particular, we show that if G excludes K t as a minor, for some fixed t ≥ 4, then col r (G) ≤ t−1 2 (2r + 1) and wcol r (G) ≤ r+t−2 t−2 · (t − 3)(2r + 1) ∈ O(r t−1 ). In the case of graphs G of bounded genus g, we improve the bounds to col r (G) ≤ (2g + 3)(2r + 1) (and even col r (G) ≤ 5r + 1 if g = 0, i.e. if G is planar) and wcol r (G) ≤ 2g + r+2 2 (2r + 1).

In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree ∆ ≥ 8 is at most 3 2 ∆ + 1. We show that it is at most 3 2 ∆ (1 + o(1)).