Nordhaus-Gaddum Theorem for the Distinguishing Chromatic Number

Nordhaus and Gaddum proved, for any graph G, that χ(G) + χ(G) 6 n + 1, where χ is the chromatic number and n = |V (G)|. Finck characterized the class of graphs, which we call NG-graphs, that satisfy equality in this bound. In this paper, we provide a new characterization of NG-graphs, based on vertex degrees, which yields a new polynomial-time recognition algorithm and efficient computation of the chromatic number of NG-graphs. Our motivation comes from our theorem that generalizes the Nordhaus-Gaddum theorem to the distinguishing chromatic number. For any graph G, χD(G) + χD(G) 6 n+D(G). We call the set of graphs that satisfy equality in this bound NGD-graphs, and characterize the set of graphs that are simultaneously NG-graphs and NGD-graphs.