On the Directed Oberwolfach Problem with variable cycle lengths

TL;DR

This paper advances the understanding of the Directed Oberwolfach Problem by establishing new factorizations of complete symmetric digraphs with variable cycle lengths for specific congruence classes of n.

Contribution

It provides new partial results on the Directed Oberwolfach Problem, including specific cycle factorizations for all n congruent to 1, 3, or 7 mod 8, and for all n ≥ 5 with cycles of length 2 and n-2.

Findings

Established a (C2,..., C2, C3)-factorization for all n ≡ 1, 3, or 7 mod 8.

Proved a (C2, C_{n-2})-factorization for all n ≥ 5.

Extended the known cases of the Directed Oberwolfach Problem.

Abstract

The Directed Oberwolfach Problem can be considered as the directed version of the well-known Oberwolfach Problem, first mentioned by Ringel at a conference in Oberwolfach, Germany in 1967. In this paper, we describe some new partial results on the Directed Oberwolfach Problem with variable cycle lengths. In particular, we show that the complete symmetric digraph $K_n^{*}$ admits a $( \vec{C}_2, ..., \vec{C}_2, \vec{C}_3) $-factorization for all $ n\equiv 1, 3,$ or $ 7\pmod{8}$. We also show that $K_n^{*}$ admits a $(\vec{C}_2, \vec{C}_{n-2})$-factorization for any integer $n \geq 5$.