(a,d)-Face antimagic graph

[math]\displaystyle{ (a,d) }[/math]-Face antimagic graph --- [math]\displaystyle{ (a,d) }[/math]-граневый антимагический граф.

A connected plane graph [math]\displaystyle{ G = (V,E,F) }[/math] is said to be [math]\displaystyle{ (a,d) }[/math]-face antimagic if there exist positive integers [math]\displaystyle{ a,b }[/math] and a bijection

[math]\displaystyle{ g: \; E(G) \rightarrow \{1,2, \ldots, |E(G)|\} }[/math]

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

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

[math]\displaystyle{ g: \; E(G) \rightarrow \{1,2, \ldots, |E(G)|\} }[/math]

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