On magicness and antimagicness of the union of 4-regular circulant graphs

Let G = (V,E) be a graph of order n and size e. An (a, d)-vertexantimagic total labeling is a bijection α from V (G) ∪ E(G) onto the set of consecutive integers {1, 2,..., n + e}, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d. The vertex-weight of a vertex x is the sum of values α(xy) assigned to all edges xy incident to the vertex x together with the value assigned to x itself. In this paper we study the vertex-magicness and vertex-antimagicness of the union of 4-regular circulant graphs.