Vertex-antimagic total labelings of graphs

In this paper we introduce a new type of graph labeling, the $(a, d)$- vertex-antimagic total labeling, which is a generalization of several other types of labelings. A connected graph $G(V,E)$ is said to be $(a, d)$-vertex-antimagic total if there exist positive integers a, d and a bijection λ : V ∪ E → \1, 2, . . . , V + E\ such that the induced mappingg_λ : V → W$ is also a bijection, where $W = \{w_λ (x) x ∈ V \} = \}a, a + d, . . . , a + ( V - 1)d$ is the set of weights of vertices in $G$. Properties of these graphs are studied. How to construct labelings for certain families of graphs are shown. Several conjectures and open problems are proposed.