Recoloring graphs of treewidth 2

Two (proper) colorings of a graph are adjacent if they differ on exactly one vertex. Jerrum proved that any $(d + 2)$-coloring of any d-degenerate graph can be transformed into any other via a sequence of adjacent colorings. A result of Bonamy et al. ensures that a shortest transformation can have a quadratic length even for $d = 1$. Bousquet and Perarnau proved that a linear transformation exists for between $(2d + 2)$-colorings. It is open to determine if this bound can be reduced. In this note, we prove that it can be reduced for graphs of treewidth 2, which are 2-degenerate. There exists a linear transformation between 5-colorings. It completes the picture for graphs of treewidth 2 since there exist graphs of treewidth 2 such a linear transformation between 4-colorings does not exist

A polynomial version of Cereceda's conjecture

Let k and d be such that k ≥ d + 2. Consider two k-colourings of a d-degenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length. The k-reconfiguration graph of G is the graph whose vertices are the proper k-colourings of G, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the (d + 2)-reconfiguration graph of any d-degenerate graph on n vertices is O(n 2). So far, the existence of a polynomial diameter is open even for d = 2. In this paper, we prove that the diameter of the k-reconfiguration graph of a d-degenerate graph is O(n d+1) for k ≥ d + 2. Moreover, we prove that if k ≥ 3 2 (d + 1) then the diameter of the k-reconfiguration graph is quadratic, improving the previous bound of k ≥ 2d + 1. We also show that the 5-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs

Fast recoloring of sparse graphs

This is a post-peer-review, pre-copyedit version of an article published in European Journal of Combinatorics. The final authenticated version is available online at: https://doi.org/10.1016/j.ejc.2015.08.001In this paper, we show that for every graph of maximum average degree bounded away from d and any two (d + 1)-colorings of it, one can transform one coloring into the other one within a polynomial number of vertex recolorings so that, at each step, the current coloring is proper. In particular, it implies that we can transform any 8-coloring of a planar graph into any other 8-coloring with a polynomial number of recolorings. These results give some evidence on a conjecture of Cereceda et al [8] which asserts that any (d + 2) coloring of a d-degenerate graph can be transformed into any other one using a polynomial number of recolorings. We also show that any (2d + 2)-coloring of a d-degenerate graph can be transformed into any other one with a linear number of recolorings.Postprint (author's final draft

Building a larger class of graphs for efficient reconfiguration of vertex colouring

A $k$-colouring of a graph $G$ is an assignment of at most $k$ colours to the vertices of $G$ so that adjacent vertices are assigned different colours. The reconfiguration graph of the $k$-colourings, $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge in $\mathcal{R}_k(G)$ if they differ in colour on exactly one vertex. For a $k$-colourable graph $G$, we investigate the connectivity and diameter of $\mathcal{R}_{k+1}(G)$. It is known that not all weakly chordal graphs have the property that $\mathcal{R}_{k+1}(G)$ is connected. On the other hand, $\mathcal{R}_{k+1}(G)$ is connected and of diameter $O(n^2)$ for several subclasses of weakly chordal graphs such as chordal, chordal bipartite, and $P_4$-free graphs. We introduce a new class of graphs called OAT graphs that extends the latter classes and in fact extends outside the class of weakly chordal graphs. OAT graphs are built from four simple operations, disjoint union, join, and the addition of a clique or comparable vertex. We prove that if $G$ is a $k$-colourable OAT graph, then $\mathcal{R}_{k+1}(G)$ is connected with diameter $O(n^2)$. Furthermore, we give polynomial time algorithms to recognize OAT graphs and to find a path between any two colourings in $\mathcal{R}_{k+1}(G)$. Feghali and Fiala defined a subclass of weakly chordal graphs, called compact graphs, and proved that for every $k$-colourable compact graph $G$, $\mathcal{R}_{k+1}(G)$ is connected with diameter $O(n^2)$. We prove that the class of OAT graphs properly contains the class of compact graphs. Feghali and Fiala also asked if for a $k$-colourable ($P_5$, co-$P_5$, $C_5$)-free graph $G$, $\mathcal{R}_{k+1}(G)$ is connected with diameter $O(n^2)$. We answer this question in the positive for the subclass of $P_4$-sparse graphs, which are the ($P_5$, co-$P_5$, $C_5$, $P$, co-$P$, fork, co-fork)-free graphs

Topics in graph colouring and extremal graph theory

In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let $G$ be a connected graph with $n$ vertices and maximum degree $\Delta(G)$. Let $R_k(G)$ denote the graph with vertex set all proper $k$-colourings of $G$ and two $k$-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that $R_{\Delta(G)+1}(G)$ has a unique non-trivial component with diameter $O(n^2)$. This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours $a$, $b$ of a component of the subgraph induced by vertices with colour $a$ or $b$. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all $\Delta(G)$-colourings of a graph $G$ are Kempe equivalent unless $G$ is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees