Abstract. For a positive integer [Formula: see text], the [Formula: see text]-recoloring graph of a graph [Formula: see text] has as vertex set all proper [Formula: see text]-colorings of [Formula: see text] with two [Formula: see text]-colorings being adjacent if they differ by the color of exactly one vertex. A result of Dyer et al. regarding graphs of bounded degeneracy implies that the 7-recoloring graphs of planar graphs, the 5-recoloring graphs of triangle-free planar graphs and the 4-recoloring graphs planar graphs of girth at least six are connected. On the other hand, there are planar graphs whose 6-recoloring graph is disconnected, triangle-free planar graphs whose 4-recoloring graph is disconnected, and planar graphs of any given girth whose 3-recoloring graph is disconnected. The main result of this paper consists in showing, via a novel application of the discharging method, that the 4-recoloring graph of every planar graph of girth five is connected. This completes the classification of the connectedness of the recoloring graph for planar graphs of given girth. We also prove some theorems regarding the diameter of the recoloring graph of planar graphs.

Under the reconfiguration framework, we consider the various ways that a target graph H is a minor of a host graph G, where a subgraph of G can be transformed into H by means of edge contraction (replacement of both endpoints of an edge by a new vertex adjacent to any vertex adjacent to either endpoint). Equivalently, an H-model of G is a labeling of the vertices of G with the vertices of H, where the contraction of all edges between identically-labeled vertices results in a graph containing representations of all edges in H. We explore the properties of G and H that result in a connected reconfiguration graph, in which nodes represent H-models and two nodes are adjacent if their corresponding H-models differ by the label of a single vertex of G. Various operations on G or H are shown to preserve connectivity. In addition, we demonstrate properties of graphs G that result in connectivity for the target graphs K_2, K_3, and K_4, including a full characterization of graphs G that result in connectivity for K_2-models, as well as the relationship between connectivity of G and other H-models.