On the Directed Oberwolfach Problem with Equal Cycle Lengths

Andrea Burgess,Mateja Šajna

doi:10.37236/2982

Andrea Burgess, Mateja Šajna

Open Access

https://doi.org/10.37236/2982

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Abstract

We examine the necessary and sufficient conditions for a complete symmetric digraph $K_n^\ast$ to admit a resolvable decomposition into directed cycles of length $m$. We give a complete solution for even $m$, and a partial solution for odd $m$.

Highlights

In this paper, we consider the problem of decomposing the complete symmetric digraphKn∗ into spanning subgraphs, each a vertex-disjoint union of directed cycles of length m
We examine the necessary and sufficient conditions for a complete symmetric digraph Kn∗ to admit a resolvable decomposition into directed cycles of length m
We consider the problem of decomposing the complete symmetric digraph

Summary

Introduction

We consider the problem of decomposing the complete symmetric digraph. We solve the problem completely for all even cycle lengths m 6, and partially for odd cycle lengths m 5, proving the following main result. There exists a resolvable decomposition of Kn∗ into directed cycles of length m if and only if m divides n and (m, n) = (6, 6). There exists a resolvable decomposition of Kn∗ into directed cycles of length m whenever n ≡ 0 (mod 4m), except possibly for n = 8m.

Preliminaries

Even cycle length

Odd cycle length

Some related results