Edge-graceful graph

Edge-graceful graph --- реберно-грациозный граф.

A graph [math]\displaystyle{ G(V,E) }[/math] is said to be edge-graceful if there exists a bijection

[math]\displaystyle{ f: \; E \rightarrow \{1,2, \ldots, |E|\} }[/math]

such that the induced mapping

[math]\displaystyle{ f^{+}: \; V \rightarrow \{0,1, \ldots, |V|-1\} }[/math]

given by

[math]\displaystyle{ f^{+}(x) = \sum\{f(xy)|xy \in E\}\pmod{|V|} }[/math]

is a bijection.

One of the well known conjectures came from Lee in 1989:

Conjecture (Lee). Every tree with an odd number of vertices is edge-graceful.

This conjecture has not been proved yet.