Optimally Reconfiguring List and Correspondence Colourings
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The reconfiguration graph 𝒞_k(G) for the k-colourings of a graph G has a vertex for each proper k-colouring of G, and two vertices of 𝒞_k(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 diam 𝒞_k(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 diam 𝒞_L(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 diam 𝒞_(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. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) diam 𝒞_L(G)≤ n(G)+2μ(G) and diam 𝒞_(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 also prove more precise results when G is a tree. 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.
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