On Nordhaus-Gaddum type relations of -complement graphs.

The -complement graphs were introduced by Amrithalakshmi et al. in 2022. In their work, some interesting properties of the graphs such as -self-complementary, adjacency, and hamiltonicity were studied. In this work, we study the coloring aspect of the -complement graphs. In particular, we provide lower and upper bounds on the product and the summation between the chromatic number and the -chromatic number of a graph, in the same fashion as the well-known Nordhaus-Gaddum type relations. Classes of graphs that achieve those bounds are also given. Furthermore, we provide upper bounds on -chromatic numbers in terms of the clique numbers and compute the -chromatic numbers of certain graphs including ladder graphs, path graphs, complete -partite graphs, and small-world Farey graphs.