Super (a,d)-edge-antimagic total labeling

Super [math]\displaystyle{ (a,d) }[/math]-edge-antimagic total labeling --- супер [math]\displaystyle{ (a,d) }[/math]-рёберно-антимагическая тотальная раскраска.

The edge-weight of an edge [math]\displaystyle{ u }[/math] under a labeling is the sum of labels (if present) carried by that edge and the vertices [math]\displaystyle{ x, y }[/math] incident with the edge [math]\displaystyle{ u }[/math].

An [math]\displaystyle{ (a, d) }[/math]edge-antimagic total labeling is defined as a bijection from [math]\displaystyle{ V(G)\bigcup E(G) }[/math] into the set [math]\displaystyle{ \{1, 2, \ldots , |V(G)|+|E(G)|\} }[/math] such that the set of edge-weights of all edges in [math]\displaystyle{ G }[/math] is equal to [math]\displaystyle{ \{a, a+d, \ldots, a+(|E|-1)d\} }[/math], for two integers [math]\displaystyle{ a \gt 0 }[/math] and [math]\displaystyle{ d \geq 0 }[/math]. An [math]\displaystyle{ (a, d) }[/math]-edge-antimagic total labeling [math]\displaystyle{ g }[/math] is called super if [math]\displaystyle{ g(V(G)) = \{1, 2, ..., |V (G)|\} }[/math] and [math]\displaystyle{ g(E(G)) = \{|V (G)| +1, |V (G)| +2, \ldots, |V (G)| + |E(G)|\} }[/math].

A graph [math]\displaystyle{ G }[/math] is called [math]\displaystyle{ (a, d) }[/math]edge-antimagic total or super [math]\displaystyle{ (a, d) }[/math]-edge-antimagic total if there exists an [math]\displaystyle{ (a, d) }[/math]-edge-antimagic total or a super [math]\displaystyle{ (a, d) }[/math]-edge-antimagic total labeling of [math]\displaystyle{ G }[/math].