Elegant labelings and Edge-colorings

A graph \(G = (V, E)\) is said to be elegant if it is possible to label its vertices by an injective mapping \(g\) into \(\{0, 1, \dots, |E|\}\) such that the induced labeling \(h\) on the edges defined for edge \(x, y\) by \(h(x, y) = g(x) + g(y) \mod (|E| + 1)\) takes all the values in \(\{1, \dots, |E|\}\). In the first part of this paper, we prove the existence of a coloring of \(K_n\) with a omnicolored path on \(n\) vertices as subgraph, which had been conjectured by Hastman [2].
In the second part we prove that the cycle on \(n\) vertices is elegant if and only if \(n \neq 1 \pmod{4}\) and we give a new construction of an elegant labeling of the path \(P_n\), where \(n \neq 4\).