Geodesic
On Super $(a, d)$-$H$-Antimagic Total Covering of Star Related Graphs
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 755-764

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Let G = (V (G), E (G)) be a simple graph and H be a subgraph of G. G admits an H-covering, if every edge in E(G) belongs to at least one subgraph of G that is isomorphic to H. An (a, d)-H-antimagic total labeling of G is a bijection λ : V (G) ∪ E(G) →1, 2, 3, . . ., |V (G)| + |E(G)| such that for all subgraphs H^' isomorphic to H, the H^′ weights wt(H^') = ∑_v ∈ V (H^') λ (v) + ∑_e ∈ E(H^')λ (e)constitute an arithmetic progression a, a+d, a+2d, . . ., a+(n−1)d where a and d are positive integers and n is the number of subgraphs of G isomorphic to H. Additionally, the labeling λ is called a super (a, d)-H-antimagic total labeling if λ (V (G)) = 1, 2, 3, . . ., |V (G)|. In this paper we study super (a, d)-H-antimagic total labelings of star related graphs G_u[S_n] and caterpillars.
Keywords: super (a, d)-H-antimagic total labeling, star 
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     title = {On {Super} $(a, d)$-$H${-Antimagic} {Total} {Covering} of {Star} {Related} {Graphs}},
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Kathiresan, K.M.; Laurence, S. David. On Super $(a, d)$-$H$-Antimagic Total Covering of Star Related Graphs. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 755-764. http://geodesic.mathdoc.fr/item/DMGT_2015_35_4_a11/