Completing the solution of the directed Oberwolfach problem with two tables

TL;DR

This paper completes the solution to the directed Oberwolfach problem with two tables of different lengths by proving the existence of specific decompositions of complete symmetric directed graphs into two directed cycles of given lengths.

Contribution

It provides a proof that the complete symmetric directed graph can be decomposed into two disjoint directed cycles of specified lengths, solving the last open case of the problem.

Findings

Proved the existence of decompositions for specific cycle lengths.

Resolved the last outstanding case of the directed Oberwolfach problem with two tables.

Combined with recent results to give a complete solution.

Abstract

We address the last outstanding case of the directed Oberwolfach problem with two tables of different lengths. Specifically, we show that the complete symmetric directed graph $K^*_n$ admits a decomposition into spanning subdigraphs comprised of two vertex-disjoint directed cycles of length $t_1$ and $t_2$, respectively, where $t_1\in \{4,6\}$, $t_2$ is even, and $t_1+t_2\geqslant 14$. In conjunction with recent results of Kadri and \v{S}ajna, this gives a complete solution to the directed Oberwolfach problem with two tables of different lengths.