On (a, d)-Total Neighborhood-Antimagic Labelings

Suppose G = (V, E) is a graph of p vertices and q edges. Let f : V ∪E → {1, 2, . . . , p+q} be a bijection such that WT(u) = Σ[ f (ux)+ f (x)] (over every neighbor x of u) is the total weight of vertex u induced by f. We say G is (a, d)-total neighborhood-antimagic if all the total weights form an arithmetic progression with first term a and common difference d. In this paper, we obtain many necessary and sufficient conditions for 1- and 2-regular graphs, and the one point union of such graphs to admit (a, d)-total neighborhood-antimagic labeling.