Edge-magic total graph

Edge-magic total graph --- реберно-магический тотальный граф.

An edge-magic total labeling on [math]\displaystyle{ G }[/math] will mean a one-to-one map [math]\displaystyle{ \lambda }[/math] from [math]\displaystyle{ V(G) \cup E(G) }[/math] onto the integers [math]\displaystyle{ 1,2, \ldots, v+e }[/math] with a property that for any edge [math]\displaystyle{ (x,y) }[/math]

[math]\displaystyle{ \lambda(x) + \lambda(x,y) + \lambda(y) = k }[/math]

for some constant [math]\displaystyle{ k }[/math]. It will be convenient to call [math]\displaystyle{ \lambda(x) + \lambda(x,y) + \lambda(y) }[/math] the edge sum of [math]\displaystyle{ (x,y) }[/math], and [math]\displaystyle{ k }[/math] (constant) a magic sum of [math]\displaystyle{ G }[/math]. A graph is called edge-magic total if it admits any edge-magic total labeling.

It is known that caterpillars and all cycles [math]\displaystyle{ C_{n} }[/math] are edge-magic total.

See also

• Vertex-magic total graph.