Super (a, d)-vertex antimagic total labeling on a disjoint union of regular graphs

摘要

Let G = (V,E) be a graph with order |G| and size |E|. An (a, d)-vertex-antimagic total labeling is a bijection a from a set of all vertices and edges to the set of consecutive integers {1,2,...,|V| + |E|}, such that the weights of the vertices form an arithmetic progression with the initial term a and the common difference d. If a(V(G)) = {1, 2,..., |V|} then we call the labeling super (a, d)-vertex antimagic total. In this paper we show some basic properties of such labelings on a disjoint union of regular graphs and show how to construct such labelings for some classes of graphs, such as cycles, generalised Pertersen graphs and circulant graphs, for d = 1.