Optimally reconfiguring list and correspondence colourings

Abstract

The reconfiguration graph Ck(G) for the k-colourings of a graph G has a vertex for each proper k-colouring of G, and two vertices of Ck(G) are adjacent precisely when those k-colourings differ on a single vertex of G. Much work has focused on bounding the maximum value of diamCk(G) over all n-vertex graphs G. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if L is a list-assignment for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamCL(G)≤n(G)+μ(G). We also conjecture that if (L,H) is a correspondence cover for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamC(L,H)(G)≤n(G)+τ(G). (Here μ(G) and τ(G) denote the matching number and vertex cover number of G.) For every graph G, we give constructions showing that both conjectures are best possible, which also hints towards an exact form of Cereceda’s Conjecture for regular graphs. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) diamCL(G)≤n(G)+2μ(G) and diamC(L,H)(G)≤n(G)+2τ(G). Our second main result proves that both conjectured bounds hold, whenever all v satisfy |L(v)|≥2d(v)+1. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree. The full paper can also be found at arxiv.org/abs/2204.07928.