A Dichotomy Theorem for Circular Colouring Reconfiguration

The "reconfiguration problem" for circular colourings asks, given two $(p,q)$-colourings $f$ and $g$ of a graph $G$, is it possible to transform $f$ into $g$ by changing the colour of one vertex at a time such that every intermediate mapping is a $(p,q)$-colouring? We show that this problem can be solved in polynomial time for $2\leq p/q <4$ and is PSPACE-complete for $p/q\geq 4$. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings.Comment: 22 pages, 5 figure

Reconfiguring Graph Homomorphisms on the Sphere

Given a loop-free graph $H$, the reconfiguration problem for homomorphisms to $H$ (also called $H$-colourings) asks: given two $H$-colourings $f$ of $g$ of a graph $G$, is it possible to transform $f$ into $g$ by a sequence of single-vertex colour changes such that every intermediate mapping is an $H$-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs $H$ (e.g. all $C_4$-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever $H$ is a $K_{2,3}$-free quadrangulation of the $2$-sphere (equivalently, the plane) which is not a $4$-cycle. From this result, we deduce an analogous statement for non-bipartite $K_{2,3}$-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and $4$-chromatic generalized Mycielski graphs, for which it was not. If we instead consider graphs $G$ and $H$ with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for $H$-colourings is PSPACE-complete whenever $H$ is a reflexive $K_{4}$-free triangulation of the $2$-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which $H$-Recolouring is known to be PSPACE-complete for reflexive instances.Comment: 22 pages, 9 figure

Characterizing Circular Colouring Mixing for pq<4\frac{p}{q}<4

Given a graph $G$, the $k$-mixing problem asks: Can one obtain all $k$-colourings of $G$, starting from one $k$-colouring $f$, by changing the colour of only one vertex at a time, while at each step maintaining a $k$-colouring? More generally, for a graph $H$, the $H$-mixing problem asks: Can one obtain all homomorphisms $G \to H$, starting from one homomorphism $f$, by changing the image of only one vertex at a time, while at each step maintaining a homomorphism $G \to H$? This paper focuses on a generalization of $k$-colourings, namely $(p,q)$-circular colourings. We show that when $2 < \frac{p}{q} < 4$, a graph $G$ is $(p,q)$-mixing if and only if for any $(p,q)$-colouring $f$ of $G$, and any cycle $C$ of $G$, the wind of the cycle under the colouring equals a particular value (which intuitively corresponds to having no wind). As a consequence we show that $(p,q)$-mixing is closed under a restricted homomorphism called a fold. Using this, we deduce that $(2k+1,k)$-mixing is co-NP-complete for all $k \in \mathbb{N}$, and by similar ideas we show that if the circular chromatic number of a connected graph $G$ is $\frac{2k+1}{k}$, then $G$ folds to $C_{2k+1}$. We use the characterization to settle a conjecture of Brewster and Noel, specifically that the circular mixing number of bipartite graphs is $2$. Lastly, we give a polynomial time algorithm for $(p,q)$-mixing in planar graphs when $3 \leq \frac{p}{q} <4$.Comment: 21 page

Graph homomorphism reconfiguration and frozen H-colourings

For a fixed graph H, the reconfiguration problem for H‐colorings (ie, homomorphisms to H) asks: given a graph G and two H‐coloring

Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs

In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs. Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber. Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses. Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture

Reconfiguring graph homomorphisms on the sphere

Given a loop-free graph H, the reconfiguration problem for homomorphisms to H (also called H-colourings) asks: given two H-colourings f of g of a graph G, is it possible to transform f into g by a sequence of single-vertex colour changes such that every intermediate mapping is an H-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs H (e.g. all C4-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever H is a K2,3-free quadrangulation of the 2-sphere (equivalently, the plane) which is not a 4-cycle. From this result, we deduce an analogous statement for non-bipartite K2,3-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and 4-chromatic generalized Mycielski graphs, for which it was not. If we instead consider graphs G and H with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for H-colourings is PSPACE-complete whenever H is a reflexive K4-free triangulation of the 2-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which H-Recolouring is known to be PSPACE-complete for reflexive instances