On the directed Oberwolfach Problem with equal cycle lengths: the odd case

TL;DR

This paper proves that complete symmetric directed graphs can be decomposed into equal-length directed cycles for all odd cycle lengths between 5 and 49, extending to larger graphs with specific divisibility conditions.

Contribution

It establishes the existence of resolvable decompositions into equal-length directed cycles for a range of odd cycle lengths in complete symmetric digraphs, covering new cases within the specified bounds.

Findings

Resolvable decompositions exist for all odd m between 5 and 49

Decompositions extend to larger graphs with n divisible by 2m

Provides constructive methods for these decompositions

Abstract

We show that the complete symmetric digraph $K_{2m}^\ast$ admits a resolvable decomposition into directed cycles of length $m$ for all odd $m$, $5 \le m \le 49$. Consequently, $K_{n}^\ast$ admits a resolvable decomposition into directed cycles of length $m$ for all $n \equiv 0 \pmod{2m}$ and odd $m$, $5 \le m \le 49$.