On Nordhaus-Gaddum type relations of δ -complement graphs.

The δ -complement graphs were introduced by Amrithalakshmi et al. in 2022. In their work, some interesting properties of the graphs such as δ -self-complementary, adjacency, and hamiltonicity were studied. In this work, we study the coloring aspect of the δ -complement graphs. In particular, we provide lower and upper bounds on the product and the summation between the chromatic number and the δ -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 δ -chromatic numbers in terms of the clique numbers and compute the δ -chromatic numbers of certain graphs including ladder graphs, path graphs, complete m -partite graphs, and small-world Farey graphs.
Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
(© 2023 The Author(s).)