Completing the solution of the directed Oberwolfach problem with cycles of equal length

TL;DR

This paper solves the last unresolved case of the directed Oberwolfach problem for tables of uniform length, proving that a complete symmetric digraph can be decomposed into directed cycles of odd length, thus settling the problem.

Contribution

It provides a complete solution to the directed Oberwolfach problem for tables of uniform length, specifically addressing the case with two tables of odd length.

Findings

Complete symmetric digraph admits a resolvable decomposition into odd-length cycles

The last outstanding case of the directed Oberwolfach problem is resolved

The problem is settled for tables of uniform length with two tables of odd size.

Abstract

In this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two-table case with tables of odd length. We prove that the complete symmetric digraph on $2m$ vertices, denoted $K^*_{2m}$, admits a resolvable decomposition into directed cycles of odd length $m$. This completely settles the directed Oberwolfach problem with tables of uniform length.