Vertex-antimagic labelings of regular graphs

Let G = (V,E) be a finite, simple and undirected graph with p vertices and q edges. An (a, d)-vertex-antimagic total labeling of G is a bijection f from V (G) ∪ E(G) onto the set of consecutive integers 1, 2, …, p + q, such that the vertex-weights form an arithmetic progression with the initial term a and difference d, where the vertex-weight of x is the sum of the value f(x) assigned to the vertex x together with all values f(xy) assigned to edges xy incident to x. Such labeling is called super if the smallest possible labels appear on the vertices.