Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k-colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for several classes of graphs and discuss important obstructions to being a coloring graph involving order, girth, and induced subgraphs.
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Beier, Julie, Janet Fierson, Ruth Haas, Heather M. Russell, and Kara Shavo. "Classifying Coloring Graphs." Discrete Mathematics 339, no. 8 (2016): 2100-2112. doi:10.1016/j.disc.2016.03.003.
Beier, Julie; Fierson, Janet; Haas, Ruth; Russell, Heather M.; and Shavo, Kara, "Classifying Coloring Graphs" (2016). Math and Computer Science Faculty Publications. 169.