A dichotomy theorem for circular colouring reconfiguration

Abstract

Let p and q be positive integers with p/q ≥ 2. 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 ≤ p/q < 4 and that it is PSPACE-complete for p/q ≥ 4. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings. As an application of the reconfiguration algorithm, we show that graphs with fewer than (k − 1)!/2 cycles of length divisible by k are k-colourable.

Introduction to Reconfiguration

“…Fixed-parameter algorithms have been found when parameterized by k + (where is the length of the reconfiguration sequence) [79,87], parameterized by k and modular-width of the input graph (and hence for cographs when parameterized by k) [75], and for shortest transformation, parameterized by k and the size of the minimum vertex cover (and hence for split graphs parameterized by k) [75]. Other variants for which reconfiguration has been studied include LIST EDGE-COLORING [36,40,100], LIST(2,1)-LABELING [37], CIRCULAR COLORING [101,102], ACYCLIC COLORING [103], and EQUITABLE COLORING [103]. The problem of k-COLORING RECONFIGURATION can also be seen as a special case of HOMOMORPHISM RECONFIGURATION [101,104] and CONSTRAINT SATISFACTION RECONFIGURATION (Section 8).…”

Introduction to Reconfiguration

“…Fixed-parameter algorithms have been found when parameterized by k + (where is the length of the reconfiguration sequence) [79,87], parameterized by k and modular-width of input graph (and hence for cographs when parameterized by k) [75], and for shortest transformation, parameterized by k and the size of the minimum vertex cover (and hence for split graphs parameterized by k) [75]. Other variants for which reconfiguration has been studied include LIST EDGE-COLORING [36,40,100], LIST(2,1)-LABELING [37], CIRCULAR COLORING [101,102], ACYCLIC COLORING [103], and EQUITABLE COLORING [103]. The problem of k-COLORING RECONFIGURATION can also be seen as a special case of HOMOMORPHISM RECONFIGURATION [101,104] and CONSTRAINT SATISFACTION RECONFIGURATION (Section 8).…”

Graph homomorphism reconfiguration and frozen H‐colorings

“…Thus, the reconfiguration problem for homomorphims to cliques admits the following dichotomy theorem. [9] generalised both the polynomial and PSPACEcomplete sides of Theorem 1.1 to circular (p, q)-cliques (see [9] for a definition): the problem is polynomial if p/q < 4 and PSPACE-complete otherwise. Intriguingly, the class of graphs for which H-Recolouring is known to be solvable in polynomial time coincides exactly with the class of graphs which are known to be multiplicative.…”

“…An important first step in each of the polynomial-time algorithms for H-Recolouring in [9,13,30] is to determine the set of vertices v of G such that ϕ ′ (v) = ϕ(v) for every homomorphism ϕ ′ which reconfigures to ϕ; i.e. to determine the set of vertices which cannot change their colour under any reconfiguration sequence starting with ϕ.…”