Let G be a simple undirected connected graph on n vertices with maximum degree . Brooks' Theorem states that G has a proper -coloring unless G is a complete graph, or a cycle with an odd number of vertices. To recolor G is to obtain a new proper coloring by changing the color of one vertex. We show an analogue of Brooks' Theorem by proving that from any k-coloring, k > , a -coloring of G can be obtained by a sequence of O(n 2 ) recolorings using only the original k colors unless -G is a complete graph or a cycle with an odd number of vertices, or -k = + 1, G is -regular and, for each vertex v in G, no two neighbors of v are colored alike.

Let G be a graph with a vertex colouring α. Let a and b be two colours. Then a connected component of the subgraph induced by those vertices coloured either a or b is known as a Kempe chain. A colouring of G obtained from α by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of G are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes.A conjecture of Mohar (2007) asserts that, for k ≥ 3, all k-colourings of a k-regular graph that is not complete are Kempe equivalent. It was later shown that all 3-colourings of a cubic graph that is neither K 4 nor the triangular prism are Kempe equivalent. In this paper, we prove that the conjecture holds for each k ≥ 4. We also report the implications of this result on the validity of the Wang-Swendsen-Kotecký algorithm for the antiferromagnetic Potts model at zero-temperature.

A graph is H-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time on (sP1 + P3)-free graphs for every integer s ≥ 1. We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. We also prove that the latter two problems are polynomial-time solvable on cographs; this was already known for Feedback Vertex Set and Odd Cycle Transversal. We complement these results by proving that Odd Cycle Transversal and Connected Odd Cycle Transversal are NP-complete on (P2 +P5, P6)-free graphs.called Connected Vertex Cover, Connected Feedback Vertex Set and Connected Odd Cycle Transversal, respectively. Garey and Johnson [15] proved that Connected Vertex Cover is NP-complete even on planar graphs of maximum degree 4 (see, for example, [14,31,36] for NPcompleteness results for other graph classes). Grigoriev and Sitters [18] proved that Connected Feedback Vertex Set is NP-complete even on planar graphs with maximum degree 9. More recently, Chiarelli et al. [10] proved that Connected Odd Cycle Transversal is NP-complete even on graphs of arbitrarily large girth and on line graphs.As all three decision problems and their connected variants are NP-complete, we can consider how to restrict the input to some special graph class in order to achieve tractability. Note that this approach is in line with the aforementioned results in the literature, where NP-completeness was proven on special graph classes. It is also in line with with, for instance, polynomial-time results for Connected Vertex Cover by Escoffier, Gourvès and Monnot [12] (for chordal graphs) and Ueno, Kajitani and Gotoh [35] (for graphs of maximum degree at most 3 and trees).Just as in most of these papers, we consider hereditary graph classes, that is, graph classes closed under vertex deletion. Hereditary graph classes form a rich framework that captures many well-studied graph classes. It is not difficult to see that every hereditary graph class G can be characterized by a (possibly infinite) set F G of forbidden induced subgraphs. If |F G | = 1, say F = {H}, then G is said to be monogenic, and every graph G ∈ G is said to be H-free. Considering monogenic graph classes can be seen as a natural first step for increasing our knowledge of the complexity of an NP-complete problem in a systematic way. Hence, we consider the following research question:How does the structure of a graph H influence the computational complexity of a graph transversal problem for input graphs that are H-free?Note that different graph transversal problems may behave differently on some class of H-free graphs. However, the general strategy for obtaining complexity results is to first try to prove that the restriction to H-free graphs is NP-complete whenever H contains a cycle or the claw (the 4-vertex star). This is usually done by showing, respectively, that the probl...

Given a graph G = (V, E) and a proper vertex colouring of G, a Kempe chain is a subset of V that induces a maximal connected subgraph of G in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for k ≥ 3, all k-colourings of k-regular graphs that are not complete are Kempe equivalent. We address the case k = 3 by showing that all 3-colourings of a cubic graph G are Kempe equivalent unless G is the complete graph K4 or the triangular prism.

The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a forest. The set A in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.

The reconfiguration graph R k (G) of the k-colourings of a graph G contains as its vertex set the k-colourings of G and two colourings are joined by an edge if they differ in colour on just one vertex of G.We show that for each k ≥ 3 there is a k-colourable weakly chordal graph G such that R k+1 (G) is disconnected. We also introduce a subclass of k-colourable weakly chordal graphs which we call k-colourable compact graphs and show that for each k-colourable compact graph G on n vertices, R k+1 (G) has diameter O(n 2 ). We show that this class contains all k-colourable co-chordal graphs and when k = 3 all 3-colourable (P 5 , P 5 , C 5 )free graphs. We also mention some open problems. *

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The reconfiguration graph R k ( G ) of the k‐colorings of a graph G has as vertex set the set of all possible k‐colorings of G and two colorings are adjacent if they differ on the color of exactly one vertex. Cereceda conjectured 10 years ago that, for every k‐degenerate graph G on n vertices, R k + 2 ( G ) has diameter scriptO ( n 2 ). The conjecture is wide open, with a best known bound of scriptO ( k n ), even for planar graphs. We improve this bound for planar graphs to 2 scriptO ( n ). Our proof can be transformed into an algorithm that runs in 2 scriptO ( n ) time.