It was not known about simple cycles
  A proper edge $t$-coloring of a graph is a coloring of its edges with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. For any integer $n\geq 3$, all possible values of $t$ are found, for which there exists such a proper edge $t$-coloring of the simple cycle C(n), which uses for each pair of adjacent edges either consecutive colors or the first and the last ones. 
R.R. Kamalian
cs.DM math.CO