(a,d)-Face antimagic graph

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(a,d)-Face antimagic graph --- (a,d)-граневый антимагический граф.

A connected plane graph G = (V,E,F) is said to be (a,d)-face antimagic if there exist positive integers a,b and a bijection

g: \; E(G) \rightarrow \{1,2, \ldots, |E(G)|\}

such that the induced mapping w_{g}^{\ast}: \; F(G) \rightarrow W is also a bijection, where W = \{w^{\ast}(f): \;f \in F(G)\} = \{a,a+d,
\ldots, a+(|F(G)| - 1)d\} is the set of weights of a face. If G =
(V,B,F) is (a,d)-face antimagic and g: \; E(G) \rightarrow \{1,2,
\ldots, |E(G)|\} is the corresponding bijective mapping of G, then g is said to be an (a,d)-face antimagic labeling of G.

The weight w^{\ast}(f) of a face f \in F(G) under an edge labeling

g: \; E(G) \rightarrow \{1,2, \ldots, |E(G)|\}

is the sum of the labels of edges surrounding that face.