Super (a, d)-vertex antimagic total labeling on a disjoint union of regular graphs * Kiki A. Sugeng, Dennl' R. Silaban Department of Mathematics, Faculty of Mathematics and Sciences, University of Indonesia Depok 16424, Indonesia {kj.ki, denny-rs}@ui. ac. id Abstract. Let G : (V,E) be a graph with order lGl and size lEl. An (a,d)-vertex-antimagic total labeling is a bijection cv from a set of all vertices and edges to the set of consecutive integers U,2,...,1V1 +lEll;, 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)) : il, 2,. . . ,lvl) 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 ofgraphs, such as cycles, generalised Pertersen graphs and circulant graphs, for d: l. 1 Introduction In this paper we consider simple and undirected graphs. The set of vertices of G will be denoted as V - V(G) and the set of edges E : Z(G), while n: lV(G)l and e : lE'(c)1. A labeli,nga of a graph G, basically is a mapping from elements (vertices, edges, faces) of a graph to set of numbers. A uerter labeli,ng (respectively, edge labeling) is a labeling which its domain is the set of vertices (respcc- tively, the set of edges). A total labeling is a Iabeling which its domain is V (C) U E(G). For a further explanation of vertex, edge and total labelings, see [a]. The aerter-weight wt(n) of a vertex r e V, under a labeling a : V U E - {1,2, ...,n * e). is c(r) * Dve N(,) o(ra), where If (r) is neighbourof z. * A part of this research is funded by "'Hibah Kompetensi Dikti 2008'' Research Grant JCMCC7l (2009), pp.2T7 -225