Vertex-antimagic total labelings of graphs
Martin Bača, François Bertault, James A. MacDougall, Mirka Miller, Rinovia Simanjuntak, Slamin Slamin · Discussiones Mathematicae Graph Theory · 2003
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 $\lambda : V \cup E \rightarrow \{1, 2, . . . , \mid V \mid + \mid E\mid\} such that the induced mapping $g_\lambda : V \rightarrow W$ is also a bijection, where $W = \{w_\lambda (x)\mid x \in V \} = \}a, a + d, . . . , a + (\mid V \mid - 1)d\mid$ 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.