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 mapping gλ : V → W is also a bijection, where W = {wλ(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.