Distance Recoloring

Coloring a graph is a well known problem and used in many different contexts. Here we want to assign k ≥ 1 colors to each vertex of a graph G such that each edge has two different colors at each endpoint. Such a vertex-coloring, if exists, is called a feasible coloring of G. Distance Coloring is an extension to the standard Coloring problem. Here we want to enforce that every pair of distinct vertices of distance less than or equal to d have different colors, for integers d ≥ 1 and k ≥ d+1. Reconfiguration problems ask if two given configurations can be transformed into each other with certain rules. For example, the well-known Coloring Reconfiguration asks if there is a way to change one vertex's color at a time, starting from a feasible given coloring α of a graph G to reach another feasible given coloring β of G, such that all intermediate colorings are also feasible. In this paper, we study the reconfiguration of distance colorings on certain graph classes. We show that even for planar, bipartite, and 2-degenerate graphs, reconfiguring distance colorings is 𝖯𝖲𝖯𝖠𝖢𝖤-complete for d ≥ 2 and k = Ω(d^2) via a reduction from the well-known Sliding Tokens problem. Additionally, we show that the problem on split graphs remains 𝖯𝖲𝖯𝖠𝖢𝖤-complete when d = 2 and large k but can be solved in polynomial time when d ≥ 3 and k ≥ d+1, and design a quadratic-time algorithm to solve the problem on paths for any d ≥ 2 and k ≥ d+1.