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

Improved hardness for H-colourings of G-colourable graphs

We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring $k$-colourable graphs with $\binom{k}{\lfloor k/2\rfloor}-1$ colours for every $k\geq 4$. This improves the result of Bul\'in, Krokhin, and Opr\v{s}al [STOC'19], who gave NP-hardness of colouring $k$-colourable graphs with $2k-1$ colours for $k\geq 3$, and the result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring $k$-colourable graphs with $2^{k^{1/3}}$ colours for sufficiently large $k$. Thus, for $k\geq 4$, we improve from known linear/sub-exponential gaps to exponential gaps. Second, we show that the topology of the box complex of H alone determines whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite, H-colourable) G. This formalises the topological intuition behind the result of Krokhin and Opr\v{s}al [FOCS'19] that 3-colouring of G-colourable graphs is NP-hard for all (3-colourable, non-bipartite) G. We use this technique to establish NP-hardness of H-colouring of G-colourable graphs for H that include but go beyond $K_3$, including square-free graphs and circular cliques (leaving $K_4$ and larger cliques open). Underlying all of our proofs is a very general observation that adjoint functors give reductions between promise constraint satisfaction problems.Comment: Mention improvement in Proposition 2.5. SODA 202

Hedetniemi’s Conjecture and Adjoint Functors in Thin Categories

We survey results on Hedetniemi’s conjecture which are connected to adjoint functors in the “thin” category of graphs, and expose the obstacles to extending these results

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

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