A procedure to obtain symmetric cycles of any odd length using directed Hajós constructions

The dichromatic number of a digraph $D$ is the minimum number of colors of a vertex coloring of $D$ such that $D$ has no monochromatic cycles. The Hajós join were recently extended to digraphs (using the dichromatic number) by J. Bang-Jensen et. al. and Hajós (directed) operations is a tool to obtain r-(di)chromatic (di)graphs. J. Bang-Jensen et. al. posed in 2020 the problem of how to obtain the symmetric cycle of length 5 from symmetric cycles of length 3. We recently solved this problem by applying a genetic algorithm. In this article, a procedure is presented to construct any odd symmetric cycle by applying directed Hajós operations to symmetric cycles of length 3, thus, generalizing the known construction of the symmetric cycle of length 5. In addition, this procedure is analyzed to determine its computational complexity.