On vertex antimagicness of disjoint union of 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. A graph which admits a (super) (a, d)-VAT labeling is said to be (super) (a, d)-VAT. Baca et al. in [2] introduced this labeling as a natural extension of the vertex-magic total labeling (VAT labeling for d = 0) defined by MacDougall et al. [3] (see also [6]). Basic properties of (a, d)-VAT labelings are investigated in [2]. In [4], it is shown how to construct super (a, d)-VAT labelings for certain families of graphs, including complete graphs, complete bipartite graphs, cycles, paths and generalized Petersen graphs. Ali et al. [1] studied properties of super (a, d)-VAT labelings and examined their existence for disjoint union of t copies of a regular graph.The idea of copies of graphs can be generalized to disjoint union of graphs. In this talk we discuss the vertex-antimagicness of disjoint union (does not have to be isomorphic) of regular graphs, especially for the case d = 1. The talk based on several results that have been published, see [5] and new results in progress.