(a,d)-Face antimagic graph

$(a,d)$-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.